TPTP Problem File: ITP148^1.p
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%------------------------------------------------------------------------------
% File : ITP148^1 : 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 : 0.50 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 466 ( 110 unt; 111 typ; 0 def)
% Number of atoms : 864 ( 450 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 2663 ( 20 ~; 2 |; 17 &;2246 @)
% ( 0 <=>; 378 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 1090 (1090 >; 0 *; 0 +; 0 <<)
% Number of symbols : 101 ( 100 usr; 10 con; 0-3 aty)
% Number of variables : 1087 ( 74 ^;1002 !; 11 ?;1087 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:42:34.421
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Complex__Ocomplex_J_J,type,
set_a_complex: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Complex__Ocomplex_Mtf__a_J_J,type,
set_complex_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
set_real_a: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (100)
thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
bij_be209634132omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001tf__a,type,
bij_betw_complex_a: ( complex > a ) > set_complex > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Real__Oreal_001t__Complex__Ocomplex,type,
bij_be122140626omplex: ( real > complex ) > set_real > set_complex > $o ).
thf(sy_c_Fun_Obij__betw_001t__Real__Oreal_001t__Real__Oreal,type,
bij_betw_real_real: ( real > real ) > set_real > set_real > $o ).
thf(sy_c_Fun_Obij__betw_001t__Real__Oreal_001tf__a,type,
bij_betw_real_a: ( real > a ) > set_real > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Complex__Ocomplex,type,
bij_betw_a_complex: ( a > complex ) > set_a > set_complex > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001tf__a,type,
bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_c130555887omplex: ( complex > complex ) > ( complex > complex ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_c595887981x_real: ( complex > complex ) > ( real > complex ) > real > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001tf__a,type,
comp_c124850173plex_a: ( complex > complex ) > ( a > complex ) > a > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
comp_c819638635l_real: ( complex > real ) > ( real > complex ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001tf__a,type,
comp_complex_real_a: ( complex > real ) > ( a > complex ) > a > real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001tf__a_001t__Complex__Ocomplex,type,
comp_c274302683omplex: ( complex > a ) > ( complex > complex ) > complex > a ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001tf__a_001t__Real__Oreal,type,
comp_complex_a_real: ( complex > a ) > ( real > complex ) > real > a ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001tf__a_001tf__a,type,
comp_complex_a_a: ( complex > a ) > ( a > complex ) > a > a ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_r667767405omplex: ( real > complex ) > ( complex > real ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_r701421291x_real: ( real > complex ) > ( real > real ) > real > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001tf__a,type,
comp_real_complex_a: ( real > complex ) > ( a > real ) > a > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Complex__Ocomplex,type,
comp_r422820971omplex: ( real > real ) > ( complex > real ) > complex > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
comp_real_real_a: ( real > real ) > ( a > real ) > a > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001t__Complex__Ocomplex,type,
comp_real_a_complex: ( real > a ) > ( complex > real ) > complex > a ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001t__Real__Oreal,type,
comp_real_a_real: ( real > a ) > ( real > real ) > real > a ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001tf__a,type,
comp_real_a_a: ( real > a ) > ( a > real ) > a > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_a1063143865omplex: ( a > complex ) > ( complex > a ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_a_complex_real: ( a > complex ) > ( real > a ) > real > complex ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Complex__Ocomplex_001tf__a,type,
comp_a_complex_a: ( a > complex ) > ( a > a ) > a > complex ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Complex__Ocomplex,type,
comp_a_real_complex: ( a > real ) > ( complex > a ) > complex > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Real__Oreal,type,
comp_a_real_real: ( a > real ) > ( real > a ) > real > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001tf__a,type,
comp_a_real_a: ( a > real ) > ( a > a ) > a > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Complex__Ocomplex,type,
comp_a_a_complex: ( a > a ) > ( complex > a ) > complex > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Real__Oreal,type,
comp_a_a_real: ( a > a ) > ( real > a ) > real > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Fun_Oid_001_062_It__Complex__Ocomplex_Mtf__a_J,type,
id_complex_a: ( complex > a ) > complex > a ).
thf(sy_c_Fun_Oid_001_062_It__Real__Oreal_Mtf__a_J,type,
id_real_a: ( real > a ) > real > a ).
thf(sy_c_Fun_Oid_001_062_Itf__a_Mtf__a_J,type,
id_a_a: ( a > a ) > a > a ).
thf(sy_c_Fun_Oid_001t__Complex__Ocomplex,type,
id_complex: complex > complex ).
thf(sy_c_Fun_Oid_001t__Real__Oreal,type,
id_real: real > real ).
thf(sy_c_Fun_Oid_001tf__a,type,
id_a: a > a ).
thf(sy_c_Fun_Oinj__on_001_062_It__Complex__Ocomplex_Mtf__a_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
inj_on893405649omplex: ( ( complex > a ) > complex > complex ) > set_complex_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Complex__Ocomplex_Mtf__a_J_001_062_It__Complex__Ocomplex_Mtf__a_J,type,
inj_on1576005937plex_a: ( ( complex > a ) > complex > a ) > set_complex_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mtf__a_J,type,
inj_on958237983real_a: ( ( real > real ) > real > a ) > set_real_real > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
inj_on319905617omplex: ( ( real > a ) > real > complex ) > set_real_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mtf__a_J,type,
inj_on_real_a_real_a: ( ( real > a ) > real > a ) > set_real_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mt__Complex__Ocomplex_J_001_062_Itf__a_Mtf__a_J,type,
inj_on_a_complex_a_a: ( ( a > complex ) > a > a ) > set_a_complex > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mt__Complex__Ocomplex_J,type,
inj_on_a_a_a_complex: ( ( a > a ) > a > complex ) > set_a_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J,type,
inj_on_a_a_a_a: ( ( a > a ) > a > a ) > set_a_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
inj_on94911183omplex: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Fun_Oinj__on_001t__Complex__Ocomplex_001tf__a,type,
inj_on_complex_a: ( complex > a ) > set_complex > $o ).
thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Complex__Ocomplex,type,
inj_on_real_complex: ( real > complex ) > set_real > $o ).
thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001tf__a,type,
inj_on_real_a: ( real > a ) > set_real > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Complex__Ocomplex,type,
inj_on_a_complex: ( a > complex ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Complex__Ocomplex_M_Eo_J,type,
top_top_complex_o: complex > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mtf__a_J_J,type,
top_to525076535plex_a: set_complex_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
top_to1446257885l_real: set_real_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
top_top_set_real_a: set_real_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mt__Complex__Ocomplex_J_J,type,
top_to2109114701omplex: set_a_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
top_top_set_a_a: set_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
top_top_set_complex: set_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Path__Connected_Oarc_001t__Complex__Ocomplex,type,
path_arc_complex: ( real > complex ) > $o ).
thf(sy_c_Path__Connected_Oarc_001t__Real__Oreal,type,
path_arc_real: ( real > real ) > $o ).
thf(sy_c_Path__Connected_Oarc_001tf__a,type,
path_arc_a: ( real > a ) > $o ).
thf(sy_c_Path__Connected_Opath_001t__Complex__Ocomplex,type,
path_path_complex: ( real > complex ) > $o ).
thf(sy_c_Path__Connected_Opath_001t__Real__Oreal,type,
path_path_real: ( real > real ) > $o ).
thf(sy_c_Path__Connected_Opath_001tf__a,type,
path_path_a: ( real > a ) > $o ).
thf(sy_c_Path__Connected_Opathfinish_001t__Complex__Ocomplex,type,
path_p769714271omplex: ( real > complex ) > complex ).
thf(sy_c_Path__Connected_Opathfinish_001t__Real__Oreal,type,
path_pathfinish_real: ( real > real ) > real ).
thf(sy_c_Path__Connected_Opathfinish_001tf__a,type,
path_pathfinish_a: ( real > a ) > a ).
thf(sy_c_Path__Connected_Opathstart_001t__Complex__Ocomplex,type,
path_p797330068omplex: ( real > complex ) > complex ).
thf(sy_c_Path__Connected_Opathstart_001t__Real__Oreal,type,
path_pathstart_real: ( real > real ) > real ).
thf(sy_c_Path__Connected_Opathstart_001tf__a,type,
path_pathstart_a: ( real > a ) > a ).
thf(sy_c_Path__Connected_Orectpath,type,
path_rectpath: complex > complex > real > complex ).
thf(sy_c_Path__Connected_Osimple__path_001t__Complex__Ocomplex,type,
path_s36253918omplex: ( real > complex ) > $o ).
thf(sy_c_Path__Connected_Osimple__path_001t__Real__Oreal,type,
path_s1005760220h_real: ( real > real ) > $o ).
thf(sy_c_Path__Connected_Osimple__path_001tf__a,type,
path_simple_path_a: ( real > a ) > $o ).
thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Ocomplex__of_001tf__a,type,
poinca1910941596x_of_a: a > complex ).
thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Oreal__of_001tf__a,type,
poinca837721858l_of_a: complex > a ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Complex__Ocomplex_001tf__a,type,
real_V762982918plex_a: ( complex > a ) > $o ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear_001tf__a_001t__Complex__Ocomplex,type,
real_V912435428omplex: ( a > complex ) > $o ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms_001t__Complex__Ocomplex_001tf__a,type,
real_V301987619plex_a: ( complex > a ) > $o ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms_001tf__a_001t__Complex__Ocomplex,type,
real_V451440129omplex: ( a > complex ) > $o ).
thf(sy_c_Real__Vector__Spaces_Olinear_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
real_V670066493omplex: ( complex > complex ) > $o ).
thf(sy_c_Real__Vector__Spaces_Olinear_001t__Complex__Ocomplex_001tf__a,type,
real_V1327653935plex_a: ( complex > a ) > $o ).
thf(sy_c_Real__Vector__Spaces_Olinear_001t__Real__Oreal_001t__Complex__Ocomplex,type,
real_V1948715323omplex: ( real > complex ) > $o ).
thf(sy_c_Real__Vector__Spaces_Olinear_001t__Real__Oreal_001t__Real__Oreal,type,
real_V1354572473l_real: ( real > real ) > $o ).
thf(sy_c_Real__Vector__Spaces_Olinear_001t__Real__Oreal_001tf__a,type,
real_V779700657real_a: ( real > a ) > $o ).
thf(sy_c_Real__Vector__Spaces_Olinear_001tf__a_001t__Complex__Ocomplex,type,
real_V1477106445omplex: ( a > complex ) > $o ).
thf(sy_c_Real__Vector__Spaces_Olinear_001tf__a_001tf__a,type,
real_V202220639ar_a_a: ( a > a ) > $o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_c,type,
c: real > a ).
% Relevant facts (354)
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_s36253918omplex @ ( comp_a_complex_real @ poinca1910941596x_of_a @ c ) ).
% a1
thf(fact_3_comp__apply,axiom,
( comp_real_a_real
= ( ^ [F: real > a,G: real > real,X: real] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_4_comp__apply,axiom,
( comp_a_complex_a
= ( ^ [F: a > complex,G: a > a,X: a] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_5_comp__apply,axiom,
( comp_a_a_complex
= ( ^ [F: a > a,G: complex > a,X: complex] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_6_comp__apply,axiom,
( comp_a_a_real
= ( ^ [F: a > a,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_7_comp__apply,axiom,
( comp_a_a_a
= ( ^ [F: a > a,G: a > a,X: a] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_8_comp__apply,axiom,
( comp_a_complex_real
= ( ^ [F: a > complex,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_9_comp__apply,axiom,
( comp_a1063143865omplex
= ( ^ [F: a > complex,G: complex > a,X: complex] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_10_comp__apply,axiom,
( comp_complex_a_a
= ( ^ [F: complex > a,G: a > complex,X: a] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_11_pathfinish__compose,axiom,
! [F2: real > a,P: real > real] :
( ( path_pathfinish_a @ ( comp_real_a_real @ F2 @ P ) )
= ( F2 @ ( path_pathfinish_real @ P ) ) ) ).
% pathfinish_compose
thf(fact_12_pathfinish__compose,axiom,
! [F2: complex > complex,P: real > complex] :
( ( path_p769714271omplex @ ( comp_c595887981x_real @ F2 @ P ) )
= ( F2 @ ( path_p769714271omplex @ P ) ) ) ).
% pathfinish_compose
thf(fact_13_pathfinish__compose,axiom,
! [F2: a > complex,P: real > a] :
( ( path_p769714271omplex @ ( comp_a_complex_real @ F2 @ P ) )
= ( F2 @ ( path_pathfinish_a @ P ) ) ) ).
% pathfinish_compose
thf(fact_14_pathfinish__compose,axiom,
! [F2: complex > a,P: real > complex] :
( ( path_pathfinish_a @ ( comp_complex_a_real @ F2 @ P ) )
= ( F2 @ ( path_p769714271omplex @ P ) ) ) ).
% pathfinish_compose
thf(fact_15_pathfinish__compose,axiom,
! [F2: a > a,P: real > a] :
( ( path_pathfinish_a @ ( comp_a_a_real @ F2 @ P ) )
= ( F2 @ ( path_pathfinish_a @ P ) ) ) ).
% pathfinish_compose
thf(fact_16_pathstart__compose,axiom,
! [F2: real > a,P: real > real] :
( ( path_pathstart_a @ ( comp_real_a_real @ F2 @ P ) )
= ( F2 @ ( path_pathstart_real @ P ) ) ) ).
% pathstart_compose
thf(fact_17_pathstart__compose,axiom,
! [F2: complex > complex,P: real > complex] :
( ( path_p797330068omplex @ ( comp_c595887981x_real @ F2 @ P ) )
= ( F2 @ ( path_p797330068omplex @ P ) ) ) ).
% pathstart_compose
thf(fact_18_pathstart__compose,axiom,
! [F2: a > complex,P: real > a] :
( ( path_p797330068omplex @ ( comp_a_complex_real @ F2 @ P ) )
= ( F2 @ ( path_pathstart_a @ P ) ) ) ).
% pathstart_compose
thf(fact_19_pathstart__compose,axiom,
! [F2: complex > a,P: real > complex] :
( ( path_pathstart_a @ ( comp_complex_a_real @ F2 @ P ) )
= ( F2 @ ( path_p797330068omplex @ P ) ) ) ).
% pathstart_compose
thf(fact_20_pathstart__compose,axiom,
! [F2: a > a,P: real > a] :
( ( path_pathstart_a @ ( comp_a_a_real @ F2 @ P ) )
= ( F2 @ ( path_pathstart_a @ P ) ) ) ).
% pathstart_compose
thf(fact_21_complex__of__bounded__linear,axiom,
real_V912435428omplex @ poinca1910941596x_of_a ).
% complex_of_bounded_linear
thf(fact_22_complex__of__linear,axiom,
real_V1477106445omplex @ poinca1910941596x_of_a ).
% complex_of_linear
thf(fact_23_comp__def,axiom,
( comp_real_a_real
= ( ^ [F: real > a,G: real > real,X: real] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_24_comp__def,axiom,
( comp_a_complex_a
= ( ^ [F: a > complex,G: a > a,X: a] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_25_comp__def,axiom,
( comp_a_a_complex
= ( ^ [F: a > a,G: complex > a,X: complex] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_26_comp__def,axiom,
( comp_a_a_real
= ( ^ [F: a > a,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_27_comp__def,axiom,
( comp_a_a_a
= ( ^ [F: a > a,G: a > a,X: a] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_28_comp__def,axiom,
( comp_a_complex_real
= ( ^ [F: a > complex,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_29_comp__def,axiom,
( comp_a1063143865omplex
= ( ^ [F: a > complex,G: complex > a,X: complex] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_30_comp__def,axiom,
( comp_complex_a_a
= ( ^ [F: complex > a,G: a > complex,X: a] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_31_comp__assoc,axiom,
! [F2: a > complex,G2: real > a,H: real > real] :
( ( comp_r701421291x_real @ ( comp_a_complex_real @ F2 @ G2 ) @ H )
= ( comp_a_complex_real @ F2 @ ( comp_real_a_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_32_comp__assoc,axiom,
! [F2: a > complex,G2: real > a,H: complex > real] :
( ( comp_r667767405omplex @ ( comp_a_complex_real @ F2 @ G2 ) @ H )
= ( comp_a1063143865omplex @ F2 @ ( comp_real_a_complex @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_33_comp__assoc,axiom,
! [F2: a > complex,G2: complex > a,H: a > complex] :
( ( comp_c124850173plex_a @ ( comp_a1063143865omplex @ F2 @ G2 ) @ H )
= ( comp_a_complex_a @ F2 @ ( comp_complex_a_a @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_34_comp__assoc,axiom,
! [F2: a > complex,G2: complex > a,H: real > complex] :
( ( comp_c595887981x_real @ ( comp_a1063143865omplex @ F2 @ G2 ) @ H )
= ( comp_a_complex_real @ F2 @ ( comp_complex_a_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_35_comp__assoc,axiom,
! [F2: a > complex,G2: complex > a,H: complex > complex] :
( ( comp_c130555887omplex @ ( comp_a1063143865omplex @ F2 @ G2 ) @ H )
= ( comp_a1063143865omplex @ F2 @ ( comp_c274302683omplex @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_36_comp__assoc,axiom,
! [F2: complex > a,G2: a > complex,H: real > a] :
( ( comp_a_a_real @ ( comp_complex_a_a @ F2 @ G2 ) @ H )
= ( comp_complex_a_real @ F2 @ ( comp_a_complex_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_37_comp__assoc,axiom,
! [F2: complex > a,G2: a > complex,H: complex > a] :
( ( comp_a_a_complex @ ( comp_complex_a_a @ F2 @ G2 ) @ H )
= ( comp_c274302683omplex @ F2 @ ( comp_a1063143865omplex @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_38_comp__assoc,axiom,
! [F2: complex > a,G2: a > complex,H: a > a] :
( ( comp_a_a_a @ ( comp_complex_a_a @ F2 @ G2 ) @ H )
= ( comp_complex_a_a @ F2 @ ( comp_a_complex_a @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_39_comp__assoc,axiom,
! [F2: complex > complex,G2: a > complex,H: real > a] :
( ( comp_a_complex_real @ ( comp_c124850173plex_a @ F2 @ G2 ) @ H )
= ( comp_c595887981x_real @ F2 @ ( comp_a_complex_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_40_comp__assoc,axiom,
! [F2: a > complex,G2: a > a,H: real > a] :
( ( comp_a_complex_real @ ( comp_a_complex_a @ F2 @ G2 ) @ H )
= ( comp_a_complex_real @ F2 @ ( comp_a_a_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_41_comp__eq__dest,axiom,
! [A: a > complex,B: real > a,C: a > complex,D: real > a,V: real] :
( ( ( comp_a_complex_real @ A @ B )
= ( comp_a_complex_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_42_comp__eq__dest,axiom,
! [A: a > complex,B: complex > a,C: a > complex,D: complex > a,V: complex] :
( ( ( comp_a1063143865omplex @ A @ B )
= ( comp_a1063143865omplex @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_43_comp__eq__dest,axiom,
! [A: complex > a,B: a > complex,C: complex > a,D: a > complex,V: a] :
( ( ( comp_complex_a_a @ A @ B )
= ( comp_complex_a_a @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_44_comp__eq__dest,axiom,
! [A: complex > a,B: a > complex,C: a > a,D: a > a,V: a] :
( ( ( comp_complex_a_a @ A @ B )
= ( comp_a_a_a @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_45_comp__eq__dest,axiom,
! [A: real > a,B: real > real,C: real > a,D: real > real,V: real] :
( ( ( comp_real_a_real @ A @ B )
= ( comp_real_a_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_46_comp__eq__dest,axiom,
! [A: real > a,B: real > real,C: a > a,D: real > a,V: real] :
( ( ( comp_real_a_real @ A @ B )
= ( comp_a_a_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_47_comp__eq__dest,axiom,
! [A: a > complex,B: a > a,C: a > complex,D: a > a,V: a] :
( ( ( comp_a_complex_a @ A @ B )
= ( comp_a_complex_a @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_48_comp__eq__dest,axiom,
! [A: a > a,B: complex > a,C: a > a,D: complex > a,V: complex] :
( ( ( comp_a_a_complex @ A @ B )
= ( comp_a_a_complex @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_49_comp__eq__dest,axiom,
! [A: a > a,B: real > a,C: real > a,D: real > real,V: real] :
( ( ( comp_a_a_real @ A @ B )
= ( comp_real_a_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_50_comp__eq__dest,axiom,
! [A: a > a,B: real > a,C: a > a,D: real > a,V: real] :
( ( ( comp_a_a_real @ A @ B )
= ( comp_a_a_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_51_comp__eq__elim,axiom,
! [A: a > complex,B: real > a,C: a > complex,D: real > a] :
( ( ( comp_a_complex_real @ A @ B )
= ( comp_a_complex_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_52_comp__eq__elim,axiom,
! [A: a > complex,B: complex > a,C: a > complex,D: complex > a] :
( ( ( comp_a1063143865omplex @ A @ B )
= ( comp_a1063143865omplex @ C @ D ) )
=> ! [V2: complex] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_53_comp__eq__elim,axiom,
! [A: complex > a,B: a > complex,C: complex > a,D: a > complex] :
( ( ( comp_complex_a_a @ A @ B )
= ( comp_complex_a_a @ C @ D ) )
=> ! [V2: a] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_54_comp__eq__elim,axiom,
! [A: complex > a,B: a > complex,C: a > a,D: a > a] :
( ( ( comp_complex_a_a @ A @ B )
= ( comp_a_a_a @ C @ D ) )
=> ! [V2: a] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_55_comp__eq__elim,axiom,
! [A: real > a,B: real > real,C: real > a,D: real > real] :
( ( ( comp_real_a_real @ A @ B )
= ( comp_real_a_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_56_comp__eq__elim,axiom,
! [A: real > a,B: real > real,C: a > a,D: real > a] :
( ( ( comp_real_a_real @ A @ B )
= ( comp_a_a_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_57_comp__eq__elim,axiom,
! [A: a > complex,B: a > a,C: a > complex,D: a > a] :
( ( ( comp_a_complex_a @ A @ B )
= ( comp_a_complex_a @ C @ D ) )
=> ! [V2: a] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_58_comp__eq__elim,axiom,
! [A: a > a,B: complex > a,C: a > a,D: complex > a] :
( ( ( comp_a_a_complex @ A @ B )
= ( comp_a_a_complex @ C @ D ) )
=> ! [V2: complex] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_59_comp__eq__elim,axiom,
! [A: a > a,B: real > a,C: real > a,D: real > real] :
( ( ( comp_a_a_real @ A @ B )
= ( comp_real_a_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_60_comp__eq__elim,axiom,
! [A: a > a,B: real > a,C: a > a,D: real > a] :
( ( ( comp_a_a_real @ A @ B )
= ( comp_a_a_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_61_comp__cong,axiom,
! [F2: a > complex,G2: real > a,X2: real,F3: a > complex,G3: real > a,X3: real] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_a_complex_real @ F2 @ G2 @ X2 )
= ( comp_a_complex_real @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_62_comp__cong,axiom,
! [F2: a > complex,G2: real > a,X2: real,F3: a > complex,G3: complex > a,X3: complex] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_a_complex_real @ F2 @ G2 @ X2 )
= ( comp_a1063143865omplex @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_63_comp__cong,axiom,
! [F2: a > complex,G2: real > a,X2: real,F3: a > complex,G3: a > a,X3: a] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_a_complex_real @ F2 @ G2 @ X2 )
= ( comp_a_complex_a @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_64_comp__cong,axiom,
! [F2: a > complex,G2: complex > a,X2: complex,F3: a > complex,G3: real > a,X3: real] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_a1063143865omplex @ F2 @ G2 @ X2 )
= ( comp_a_complex_real @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_65_comp__cong,axiom,
! [F2: a > complex,G2: complex > a,X2: complex,F3: a > complex,G3: complex > a,X3: complex] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_a1063143865omplex @ F2 @ G2 @ X2 )
= ( comp_a1063143865omplex @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_66_comp__cong,axiom,
! [F2: a > complex,G2: complex > a,X2: complex,F3: a > complex,G3: a > a,X3: a] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_a1063143865omplex @ F2 @ G2 @ X2 )
= ( comp_a_complex_a @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_67_comp__cong,axiom,
! [F2: complex > a,G2: a > complex,X2: a,F3: complex > a,G3: a > complex,X3: a] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_complex_a_a @ F2 @ G2 @ X2 )
= ( comp_complex_a_a @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_68_comp__cong,axiom,
! [F2: complex > a,G2: a > complex,X2: a,F3: real > a,G3: real > real,X3: real] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_complex_a_a @ F2 @ G2 @ X2 )
= ( comp_real_a_real @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_69_comp__cong,axiom,
! [F2: complex > a,G2: a > complex,X2: a,F3: a > a,G3: complex > a,X3: complex] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_complex_a_a @ F2 @ G2 @ X2 )
= ( comp_a_a_complex @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_70_comp__cong,axiom,
! [F2: complex > a,G2: a > complex,X2: a,F3: a > a,G3: real > a,X3: real] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_complex_a_a @ F2 @ G2 @ X2 )
= ( comp_a_a_real @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_71_comp__eq__dest__lhs,axiom,
! [A: a > complex,B: real > a,C: real > complex,V: real] :
( ( ( comp_a_complex_real @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_72_comp__eq__dest__lhs,axiom,
! [A: a > complex,B: complex > a,C: complex > complex,V: complex] :
( ( ( comp_a1063143865omplex @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_73_comp__eq__dest__lhs,axiom,
! [A: complex > a,B: a > complex,C: a > a,V: a] :
( ( ( comp_complex_a_a @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_74_comp__eq__dest__lhs,axiom,
! [A: real > a,B: real > real,C: real > a,V: real] :
( ( ( comp_real_a_real @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_75_comp__eq__dest__lhs,axiom,
! [A: a > complex,B: a > a,C: a > complex,V: a] :
( ( ( comp_a_complex_a @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_76_comp__eq__dest__lhs,axiom,
! [A: a > a,B: complex > a,C: complex > a,V: complex] :
( ( ( comp_a_a_complex @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_77_comp__eq__dest__lhs,axiom,
! [A: a > a,B: real > a,C: real > a,V: real] :
( ( ( comp_a_a_real @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_78_comp__eq__dest__lhs,axiom,
! [A: a > a,B: a > a,C: a > a,V: a] :
( ( ( comp_a_a_a @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_79_pathstart__linear__image__eq,axiom,
! [F2: complex > complex,G2: real > complex] :
( ( real_V670066493omplex @ F2 )
=> ( ( path_p797330068omplex @ ( comp_c595887981x_real @ F2 @ G2 ) )
= ( F2 @ ( path_p797330068omplex @ G2 ) ) ) ) ).
% pathstart_linear_image_eq
thf(fact_80_pathstart__linear__image__eq,axiom,
! [F2: real > a,G2: real > real] :
( ( real_V779700657real_a @ F2 )
=> ( ( path_pathstart_a @ ( comp_real_a_real @ F2 @ G2 ) )
= ( F2 @ ( path_pathstart_real @ G2 ) ) ) ) ).
% pathstart_linear_image_eq
thf(fact_81_pathstart__linear__image__eq,axiom,
! [F2: a > a,G2: real > a] :
( ( real_V202220639ar_a_a @ F2 )
=> ( ( path_pathstart_a @ ( comp_a_a_real @ F2 @ G2 ) )
= ( F2 @ ( path_pathstart_a @ G2 ) ) ) ) ).
% pathstart_linear_image_eq
thf(fact_82_pathstart__linear__image__eq,axiom,
! [F2: a > complex,G2: real > a] :
( ( real_V1477106445omplex @ F2 )
=> ( ( path_p797330068omplex @ ( comp_a_complex_real @ F2 @ G2 ) )
= ( F2 @ ( path_pathstart_a @ G2 ) ) ) ) ).
% pathstart_linear_image_eq
thf(fact_83_pathstart__linear__image__eq,axiom,
! [F2: complex > a,G2: real > complex] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( path_pathstart_a @ ( comp_complex_a_real @ F2 @ G2 ) )
= ( F2 @ ( path_p797330068omplex @ G2 ) ) ) ) ).
% pathstart_linear_image_eq
thf(fact_84_pathfinish__linear__image,axiom,
! [F2: complex > complex,G2: real > complex] :
( ( real_V670066493omplex @ F2 )
=> ( ( path_p769714271omplex @ ( comp_c595887981x_real @ F2 @ G2 ) )
= ( F2 @ ( path_p769714271omplex @ G2 ) ) ) ) ).
% pathfinish_linear_image
thf(fact_85_pathfinish__linear__image,axiom,
! [F2: real > a,G2: real > real] :
( ( real_V779700657real_a @ F2 )
=> ( ( path_pathfinish_a @ ( comp_real_a_real @ F2 @ G2 ) )
= ( F2 @ ( path_pathfinish_real @ G2 ) ) ) ) ).
% pathfinish_linear_image
thf(fact_86_pathfinish__linear__image,axiom,
! [F2: a > a,G2: real > a] :
( ( real_V202220639ar_a_a @ F2 )
=> ( ( path_pathfinish_a @ ( comp_a_a_real @ F2 @ G2 ) )
= ( F2 @ ( path_pathfinish_a @ G2 ) ) ) ) ).
% pathfinish_linear_image
thf(fact_87_pathfinish__linear__image,axiom,
! [F2: a > complex,G2: real > a] :
( ( real_V1477106445omplex @ F2 )
=> ( ( path_p769714271omplex @ ( comp_a_complex_real @ F2 @ G2 ) )
= ( F2 @ ( path_pathfinish_a @ G2 ) ) ) ) ).
% pathfinish_linear_image
thf(fact_88_pathfinish__linear__image,axiom,
! [F2: complex > a,G2: real > complex] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( path_pathfinish_a @ ( comp_complex_a_real @ F2 @ G2 ) )
= ( F2 @ ( path_p769714271omplex @ G2 ) ) ) ) ).
% pathfinish_linear_image
thf(fact_89_bounded__linear_Olinear,axiom,
! [F2: a > complex] :
( ( real_V912435428omplex @ F2 )
=> ( real_V1477106445omplex @ F2 ) ) ).
% bounded_linear.linear
thf(fact_90_bounded__linear_Olinear,axiom,
! [F2: complex > a] :
( ( real_V762982918plex_a @ F2 )
=> ( real_V1327653935plex_a @ F2 ) ) ).
% bounded_linear.linear
thf(fact_91_linear__conv__bounded__linear,axiom,
real_V1477106445omplex = real_V912435428omplex ).
% linear_conv_bounded_linear
thf(fact_92_linear__conv__bounded__linear,axiom,
real_V1327653935plex_a = real_V762982918plex_a ).
% linear_conv_bounded_linear
thf(fact_93_linear__compose,axiom,
! [F2: real > real,G2: real > a] :
( ( real_V1354572473l_real @ F2 )
=> ( ( real_V779700657real_a @ G2 )
=> ( real_V779700657real_a @ ( comp_real_a_real @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_94_linear__compose,axiom,
! [F2: real > a,G2: a > a] :
( ( real_V779700657real_a @ F2 )
=> ( ( real_V202220639ar_a_a @ G2 )
=> ( real_V779700657real_a @ ( comp_a_a_real @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_95_linear__compose,axiom,
! [F2: a > a,G2: a > a] :
( ( real_V202220639ar_a_a @ F2 )
=> ( ( real_V202220639ar_a_a @ G2 )
=> ( real_V202220639ar_a_a @ ( comp_a_a_a @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_96_linear__compose,axiom,
! [F2: real > a,G2: a > complex] :
( ( real_V779700657real_a @ F2 )
=> ( ( real_V1477106445omplex @ G2 )
=> ( real_V1948715323omplex @ ( comp_a_complex_real @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_97_linear__compose,axiom,
! [F2: a > a,G2: a > complex] :
( ( real_V202220639ar_a_a @ F2 )
=> ( ( real_V1477106445omplex @ G2 )
=> ( real_V1477106445omplex @ ( comp_a_complex_a @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_98_linear__compose,axiom,
! [F2: complex > complex,G2: complex > a] :
( ( real_V670066493omplex @ F2 )
=> ( ( real_V1327653935plex_a @ G2 )
=> ( real_V1327653935plex_a @ ( comp_c274302683omplex @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_99_linear__compose,axiom,
! [F2: a > complex,G2: complex > complex] :
( ( real_V1477106445omplex @ F2 )
=> ( ( real_V670066493omplex @ G2 )
=> ( real_V1477106445omplex @ ( comp_c124850173plex_a @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_100_linear__compose,axiom,
! [F2: a > complex,G2: complex > a] :
( ( real_V1477106445omplex @ F2 )
=> ( ( real_V1327653935plex_a @ G2 )
=> ( real_V202220639ar_a_a @ ( comp_complex_a_a @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_101_linear__compose,axiom,
! [F2: complex > a,G2: a > a] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( real_V202220639ar_a_a @ G2 )
=> ( real_V1327653935plex_a @ ( comp_a_a_complex @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_102_linear__compose,axiom,
! [F2: complex > a,G2: a > complex] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( real_V1477106445omplex @ G2 )
=> ( real_V670066493omplex @ ( comp_a1063143865omplex @ G2 @ F2 ) ) ) ) ).
% linear_compose
thf(fact_103__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 @ poinca1910941596x_of_a @ top_top_set_a )
=> ( ( path_s36253918omplex @ ( comp_a_complex_real @ poinca1910941596x_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_104_pathfinish__rectpath,axiom,
! [A1: complex,A3: complex] :
( ( path_p769714271omplex @ ( path_rectpath @ A1 @ A3 ) )
= A1 ) ).
% pathfinish_rectpath
thf(fact_105_pathstart__rectpath,axiom,
! [A1: complex,A3: complex] :
( ( path_p797330068omplex @ ( path_rectpath @ A1 @ A3 ) )
= A1 ) ).
% pathstart_rectpath
thf(fact_106_bounded__linear_Obounded__linear,axiom,
! [F2: a > complex] :
( ( real_V912435428omplex @ F2 )
=> ( real_V912435428omplex @ F2 ) ) ).
% bounded_linear.bounded_linear
thf(fact_107_bounded__linear_Obounded__linear,axiom,
! [F2: complex > a] :
( ( real_V762982918plex_a @ F2 )
=> ( real_V762982918plex_a @ F2 ) ) ).
% bounded_linear.bounded_linear
thf(fact_108_rewriteR__comp__comp2,axiom,
! [G2: real > real,H: a > real,R1: complex > real,R2: a > complex,F2: real > a,L: complex > a] :
( ( ( comp_real_real_a @ G2 @ H )
= ( comp_complex_real_a @ R1 @ R2 ) )
=> ( ( ( comp_real_a_complex @ F2 @ R1 )
= L )
=> ( ( comp_real_a_a @ ( comp_real_a_real @ F2 @ G2 ) @ H )
= ( comp_complex_a_a @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_109_rewriteR__comp__comp2,axiom,
! [G2: real > real,H: complex > real,R1: a > real,R2: complex > a,F2: real > a,L: a > a] :
( ( ( comp_r422820971omplex @ G2 @ H )
= ( comp_a_real_complex @ R1 @ R2 ) )
=> ( ( ( comp_real_a_a @ F2 @ R1 )
= L )
=> ( ( comp_real_a_complex @ ( comp_real_a_real @ F2 @ G2 ) @ H )
= ( comp_a_a_complex @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_110_rewriteR__comp__comp2,axiom,
! [G2: real > real,H: a > real,R1: a > real,R2: a > a,F2: real > a,L: a > a] :
( ( ( comp_real_real_a @ G2 @ H )
= ( comp_a_real_a @ R1 @ R2 ) )
=> ( ( ( comp_real_a_a @ F2 @ R1 )
= L )
=> ( ( comp_real_a_a @ ( comp_real_a_real @ F2 @ G2 ) @ H )
= ( comp_a_a_a @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_111_rewriteR__comp__comp2,axiom,
! [G2: real > real,H: real > real,R1: a > real,R2: real > a,F2: real > a,L: a > a] :
( ( ( comp_real_real_real @ G2 @ H )
= ( comp_a_real_real @ R1 @ R2 ) )
=> ( ( ( comp_real_a_a @ F2 @ R1 )
= L )
=> ( ( comp_real_a_real @ ( comp_real_a_real @ F2 @ G2 ) @ H )
= ( comp_a_a_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_112_rewriteR__comp__comp2,axiom,
! [G2: complex > real,H: a > complex,R1: real > real,R2: a > real,F2: real > a,L: real > a] :
( ( ( comp_complex_real_a @ G2 @ H )
= ( comp_real_real_a @ R1 @ R2 ) )
=> ( ( ( comp_real_a_real @ F2 @ R1 )
= L )
=> ( ( comp_complex_a_a @ ( comp_real_a_complex @ F2 @ G2 ) @ H )
= ( comp_real_a_a @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_113_rewriteR__comp__comp2,axiom,
! [G2: real > real,H: real > real,R1: real > real,R2: real > real,F2: real > a,L: real > a] :
( ( ( comp_real_real_real @ G2 @ H )
= ( comp_real_real_real @ R1 @ R2 ) )
=> ( ( ( comp_real_a_real @ F2 @ R1 )
= L )
=> ( ( comp_real_a_real @ ( comp_real_a_real @ F2 @ G2 ) @ H )
= ( comp_real_a_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_114_rewriteR__comp__comp2,axiom,
! [G2: a > real,H: complex > a,R1: real > real,R2: complex > real,F2: real > a,L: real > a] :
( ( ( comp_a_real_complex @ G2 @ H )
= ( comp_r422820971omplex @ R1 @ R2 ) )
=> ( ( ( comp_real_a_real @ F2 @ R1 )
= L )
=> ( ( comp_a_a_complex @ ( comp_real_a_a @ F2 @ G2 ) @ H )
= ( comp_real_a_complex @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_115_rewriteR__comp__comp2,axiom,
! [G2: a > real,H: real > a,R1: real > real,R2: real > real,F2: real > a,L: real > a] :
( ( ( comp_a_real_real @ G2 @ H )
= ( comp_real_real_real @ R1 @ R2 ) )
=> ( ( ( comp_real_a_real @ F2 @ R1 )
= L )
=> ( ( comp_a_a_real @ ( comp_real_a_a @ F2 @ G2 ) @ H )
= ( comp_real_a_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_116_rewriteR__comp__comp2,axiom,
! [G2: a > real,H: a > a,R1: real > real,R2: a > real,F2: real > a,L: real > a] :
( ( ( comp_a_real_a @ G2 @ H )
= ( comp_real_real_a @ R1 @ R2 ) )
=> ( ( ( comp_real_a_real @ F2 @ R1 )
= L )
=> ( ( comp_a_a_a @ ( comp_real_a_a @ F2 @ G2 ) @ H )
= ( comp_real_a_a @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_117_rewriteR__comp__comp2,axiom,
! [G2: real > complex,H: real > real,R1: a > complex,R2: real > a,F2: complex > a,L: a > a] :
( ( ( comp_r701421291x_real @ G2 @ H )
= ( comp_a_complex_real @ R1 @ R2 ) )
=> ( ( ( comp_complex_a_a @ F2 @ R1 )
= L )
=> ( ( comp_real_a_real @ ( comp_complex_a_real @ F2 @ G2 ) @ H )
= ( comp_a_a_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_118_rewriteL__comp__comp2,axiom,
! [F2: real > a,G2: a > real,L1: complex > a,L2: a > complex,H: real > a,R: real > complex] :
( ( ( comp_real_a_a @ F2 @ G2 )
= ( comp_complex_a_a @ L1 @ L2 ) )
=> ( ( ( comp_a_complex_real @ L2 @ H )
= R )
=> ( ( comp_real_a_real @ F2 @ ( comp_a_real_real @ G2 @ H ) )
= ( comp_complex_a_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_119_rewriteL__comp__comp2,axiom,
! [F2: complex > a,G2: real > complex,L1: real > a,L2: real > real,H: a > real,R: a > real] :
( ( ( comp_complex_a_real @ F2 @ G2 )
= ( comp_real_a_real @ L1 @ L2 ) )
=> ( ( ( comp_real_real_a @ L2 @ H )
= R )
=> ( ( comp_complex_a_a @ F2 @ ( comp_real_complex_a @ G2 @ H ) )
= ( comp_real_a_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_120_rewriteL__comp__comp2,axiom,
! [F2: complex > complex,G2: a > complex,L1: a > complex,L2: a > a,H: complex > a,R: complex > a] :
( ( ( comp_c124850173plex_a @ F2 @ G2 )
= ( comp_a_complex_a @ L1 @ L2 ) )
=> ( ( ( comp_a_a_complex @ L2 @ H )
= R )
=> ( ( comp_c130555887omplex @ F2 @ ( comp_a1063143865omplex @ G2 @ H ) )
= ( comp_a1063143865omplex @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_121_rewriteL__comp__comp2,axiom,
! [F2: complex > complex,G2: a > complex,L1: a > complex,L2: a > a,H: real > a,R: real > a] :
( ( ( comp_c124850173plex_a @ F2 @ G2 )
= ( comp_a_complex_a @ L1 @ L2 ) )
=> ( ( ( comp_a_a_real @ L2 @ H )
= R )
=> ( ( comp_c595887981x_real @ F2 @ ( comp_a_complex_real @ G2 @ H ) )
= ( comp_a_complex_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_122_rewriteL__comp__comp2,axiom,
! [F2: complex > complex,G2: a > complex,L1: a > complex,L2: a > a,H: a > a,R: a > a] :
( ( ( comp_c124850173plex_a @ F2 @ G2 )
= ( comp_a_complex_a @ L1 @ L2 ) )
=> ( ( ( comp_a_a_a @ L2 @ H )
= R )
=> ( ( comp_c124850173plex_a @ F2 @ ( comp_a_complex_a @ G2 @ H ) )
= ( comp_a_complex_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_123_rewriteL__comp__comp2,axiom,
! [F2: real > a,G2: complex > real,L1: a > a,L2: complex > a,H: real > complex,R: real > a] :
( ( ( comp_real_a_complex @ F2 @ G2 )
= ( comp_a_a_complex @ L1 @ L2 ) )
=> ( ( ( comp_complex_a_real @ L2 @ H )
= R )
=> ( ( comp_real_a_real @ F2 @ ( comp_c819638635l_real @ G2 @ H ) )
= ( comp_a_a_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_124_rewriteL__comp__comp2,axiom,
! [F2: complex > a,G2: complex > complex,L1: a > a,L2: complex > a,H: a > complex,R: a > a] :
( ( ( comp_c274302683omplex @ F2 @ G2 )
= ( comp_a_a_complex @ L1 @ L2 ) )
=> ( ( ( comp_complex_a_a @ L2 @ H )
= R )
=> ( ( comp_complex_a_a @ F2 @ ( comp_c124850173plex_a @ G2 @ H ) )
= ( comp_a_a_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_125_rewriteL__comp__comp2,axiom,
! [F2: complex > a,G2: real > complex,L1: a > a,L2: real > a,H: a > real,R: a > a] :
( ( ( comp_complex_a_real @ F2 @ G2 )
= ( comp_a_a_real @ L1 @ L2 ) )
=> ( ( ( comp_real_a_a @ L2 @ H )
= R )
=> ( ( comp_complex_a_a @ F2 @ ( comp_real_complex_a @ G2 @ H ) )
= ( comp_a_a_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_126_rewriteL__comp__comp2,axiom,
! [F2: real > a,G2: a > real,L1: a > a,L2: a > a,H: real > a,R: real > a] :
( ( ( comp_real_a_a @ F2 @ G2 )
= ( comp_a_a_a @ L1 @ L2 ) )
=> ( ( ( comp_a_a_real @ L2 @ H )
= R )
=> ( ( comp_real_a_real @ F2 @ ( comp_a_real_real @ G2 @ H ) )
= ( comp_a_a_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_127_rewriteL__comp__comp2,axiom,
! [F2: a > complex,G2: real > a,L1: a > complex,L2: real > a,H: complex > real,R: complex > a] :
( ( ( comp_a_complex_real @ F2 @ G2 )
= ( comp_a_complex_real @ L1 @ L2 ) )
=> ( ( ( comp_real_a_complex @ L2 @ H )
= R )
=> ( ( comp_a1063143865omplex @ F2 @ ( comp_real_a_complex @ G2 @ H ) )
= ( comp_a1063143865omplex @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_128_rewriteR__comp__comp,axiom,
! [G2: real > a,H: complex > real,R: complex > a,F2: a > complex] :
( ( ( comp_real_a_complex @ G2 @ H )
= R )
=> ( ( comp_r667767405omplex @ ( comp_a_complex_real @ F2 @ G2 ) @ H )
= ( comp_a1063143865omplex @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_129_rewriteR__comp__comp,axiom,
! [G2: real > a,H: a > real,R: a > a,F2: a > complex] :
( ( ( comp_real_a_a @ G2 @ H )
= R )
=> ( ( comp_real_complex_a @ ( comp_a_complex_real @ F2 @ G2 ) @ H )
= ( comp_a_complex_a @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_130_rewriteR__comp__comp,axiom,
! [G2: complex > a,H: real > complex,R: real > a,F2: a > complex] :
( ( ( comp_complex_a_real @ G2 @ H )
= R )
=> ( ( comp_c595887981x_real @ ( comp_a1063143865omplex @ F2 @ G2 ) @ H )
= ( comp_a_complex_real @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_131_rewriteR__comp__comp,axiom,
! [G2: complex > a,H: complex > complex,R: complex > a,F2: a > complex] :
( ( ( comp_c274302683omplex @ G2 @ H )
= R )
=> ( ( comp_c130555887omplex @ ( comp_a1063143865omplex @ F2 @ G2 ) @ H )
= ( comp_a1063143865omplex @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_132_rewriteR__comp__comp,axiom,
! [G2: complex > a,H: complex > complex,R: complex > a,F2: a > a] :
( ( ( comp_c274302683omplex @ G2 @ H )
= R )
=> ( ( comp_c274302683omplex @ ( comp_a_a_complex @ F2 @ G2 ) @ H )
= ( comp_a_a_complex @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_133_rewriteR__comp__comp,axiom,
! [G2: complex > a,H: real > complex,R: real > a,F2: a > a] :
( ( ( comp_complex_a_real @ G2 @ H )
= R )
=> ( ( comp_complex_a_real @ ( comp_a_a_complex @ F2 @ G2 ) @ H )
= ( comp_a_a_real @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_134_rewriteR__comp__comp,axiom,
! [G2: real > a,H: complex > real,R: complex > a,F2: a > a] :
( ( ( comp_real_a_complex @ G2 @ H )
= R )
=> ( ( comp_real_a_complex @ ( comp_a_a_real @ F2 @ G2 ) @ H )
= ( comp_a_a_complex @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_135_rewriteR__comp__comp,axiom,
! [G2: real > a,H: a > real,R: a > a,F2: a > a] :
( ( ( comp_real_a_a @ G2 @ H )
= R )
=> ( ( comp_real_a_a @ ( comp_a_a_real @ F2 @ G2 ) @ H )
= ( comp_a_a_a @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_136_rewriteR__comp__comp,axiom,
! [G2: complex > complex,H: a > complex,R: a > complex,F2: complex > a] :
( ( ( comp_c124850173plex_a @ G2 @ H )
= R )
=> ( ( comp_complex_a_a @ ( comp_c274302683omplex @ F2 @ G2 ) @ H )
= ( comp_complex_a_a @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_137_rewriteR__comp__comp,axiom,
! [G2: real > real,H: real > real,R: real > real,F2: real > a] :
( ( ( comp_real_real_real @ G2 @ H )
= R )
=> ( ( comp_real_a_real @ ( comp_real_a_real @ F2 @ G2 ) @ H )
= ( comp_real_a_real @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_138_rewriteL__comp__comp,axiom,
! [F2: complex > complex,G2: a > complex,L: a > complex,H: real > a] :
( ( ( comp_c124850173plex_a @ F2 @ G2 )
= L )
=> ( ( comp_c595887981x_real @ F2 @ ( comp_a_complex_real @ G2 @ H ) )
= ( comp_a_complex_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_139_rewriteL__comp__comp,axiom,
! [F2: complex > complex,G2: a > complex,L: a > complex,H: complex > a] :
( ( ( comp_c124850173plex_a @ F2 @ G2 )
= L )
=> ( ( comp_c130555887omplex @ F2 @ ( comp_a1063143865omplex @ G2 @ H ) )
= ( comp_a1063143865omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_140_rewriteL__comp__comp,axiom,
! [F2: complex > complex,G2: a > complex,L: a > complex,H: a > a] :
( ( ( comp_c124850173plex_a @ F2 @ G2 )
= L )
=> ( ( comp_c124850173plex_a @ F2 @ ( comp_a_complex_a @ G2 @ H ) )
= ( comp_a_complex_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_141_rewriteL__comp__comp,axiom,
! [F2: complex > a,G2: complex > complex,L: complex > a,H: a > complex] :
( ( ( comp_c274302683omplex @ F2 @ G2 )
= L )
=> ( ( comp_complex_a_a @ F2 @ ( comp_c124850173plex_a @ G2 @ H ) )
= ( comp_complex_a_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_142_rewriteL__comp__comp,axiom,
! [F2: real > a,G2: a > real,L: a > a,H: real > a] :
( ( ( comp_real_a_a @ F2 @ G2 )
= L )
=> ( ( comp_real_a_real @ F2 @ ( comp_a_real_real @ G2 @ H ) )
= ( comp_a_a_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_143_rewriteL__comp__comp,axiom,
! [F2: a > complex,G2: real > a,L: real > complex,H: real > real] :
( ( ( comp_a_complex_real @ F2 @ G2 )
= L )
=> ( ( comp_a_complex_real @ F2 @ ( comp_real_a_real @ G2 @ H ) )
= ( comp_r701421291x_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_144_rewriteL__comp__comp,axiom,
! [F2: a > complex,G2: real > a,L: real > complex,H: complex > real] :
( ( ( comp_a_complex_real @ F2 @ G2 )
= L )
=> ( ( comp_a1063143865omplex @ F2 @ ( comp_real_a_complex @ G2 @ H ) )
= ( comp_r667767405omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_145_rewriteL__comp__comp,axiom,
! [F2: a > complex,G2: real > a,L: real > complex,H: a > real] :
( ( ( comp_a_complex_real @ F2 @ G2 )
= L )
=> ( ( comp_a_complex_a @ F2 @ ( comp_real_a_a @ G2 @ H ) )
= ( comp_real_complex_a @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_146_rewriteL__comp__comp,axiom,
! [F2: a > complex,G2: complex > a,L: complex > complex,H: real > complex] :
( ( ( comp_a1063143865omplex @ F2 @ G2 )
= L )
=> ( ( comp_a_complex_real @ F2 @ ( comp_complex_a_real @ G2 @ H ) )
= ( comp_c595887981x_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_147_rewriteL__comp__comp,axiom,
! [F2: a > complex,G2: complex > a,L: complex > complex,H: complex > complex] :
( ( ( comp_a1063143865omplex @ F2 @ G2 )
= L )
=> ( ( comp_a1063143865omplex @ F2 @ ( comp_c274302683omplex @ G2 @ H ) )
= ( comp_c130555887omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_148_injD,axiom,
! [F2: a > complex,X2: a,Y: a] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( X2 = Y ) ) ) ).
% injD
thf(fact_149_injD,axiom,
! [F2: complex > a,X2: complex,Y: complex] :
( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( X2 = Y ) ) ) ).
% injD
thf(fact_150_injI,axiom,
! [F2: a > complex] :
( ! [X4: a,Y2: a] :
( ( ( F2 @ X4 )
= ( F2 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on_a_complex @ F2 @ top_top_set_a ) ) ).
% injI
thf(fact_151_injI,axiom,
! [F2: complex > a] :
( ! [X4: complex,Y2: complex] :
( ( ( F2 @ X4 )
= ( F2 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on_complex_a @ F2 @ top_top_set_complex ) ) ).
% injI
thf(fact_152_inj__eq,axiom,
! [F2: a > complex,X2: a,Y: a] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
= ( X2 = Y ) ) ) ).
% inj_eq
thf(fact_153_inj__eq,axiom,
! [F2: complex > a,X2: complex,Y: complex] :
( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
= ( X2 = Y ) ) ) ).
% inj_eq
thf(fact_154_inj__def,axiom,
! [F2: a > complex] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
= ( ! [X: a,Y3: a] :
( ( ( F2 @ X )
= ( F2 @ Y3 ) )
=> ( X = Y3 ) ) ) ) ).
% inj_def
thf(fact_155_inj__def,axiom,
! [F2: complex > a] :
( ( inj_on_complex_a @ F2 @ top_top_set_complex )
= ( ! [X: complex,Y3: complex] :
( ( ( F2 @ X )
= ( F2 @ Y3 ) )
=> ( X = Y3 ) ) ) ) ).
% inj_def
thf(fact_156_inj__onD,axiom,
! [F2: a > complex,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_complex @ F2 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_157_inj__onD,axiom,
! [F2: complex > a,A2: set_complex,X2: complex,Y: complex] :
( ( inj_on_complex_a @ F2 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( ( member_complex @ X2 @ A2 )
=> ( ( member_complex @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_158_inj__onI,axiom,
! [A2: set_a,F2: a > complex] :
( ! [X4: a,Y2: a] :
( ( member_a @ X4 @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( ( ( F2 @ X4 )
= ( F2 @ Y2 ) )
=> ( X4 = Y2 ) ) ) )
=> ( inj_on_a_complex @ F2 @ A2 ) ) ).
% inj_onI
thf(fact_159_inj__onI,axiom,
! [A2: set_complex,F2: complex > a] :
( ! [X4: complex,Y2: complex] :
( ( member_complex @ X4 @ A2 )
=> ( ( member_complex @ Y2 @ A2 )
=> ( ( ( F2 @ X4 )
= ( F2 @ Y2 ) )
=> ( X4 = Y2 ) ) ) )
=> ( inj_on_complex_a @ F2 @ A2 ) ) ).
% inj_onI
thf(fact_160_inj__on__def,axiom,
( inj_on_a_complex
= ( ^ [F: a > complex,A4: set_a] :
! [X: a] :
( ( member_a @ X @ A4 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ A4 )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_161_inj__on__def,axiom,
( inj_on_complex_a
= ( ^ [F: complex > a,A4: set_complex] :
! [X: complex] :
( ( member_complex @ X @ A4 )
=> ! [Y3: complex] :
( ( member_complex @ Y3 @ A4 )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_162_inj__on__cong,axiom,
! [A2: set_a,F2: a > complex,G2: a > complex] :
( ! [A5: a] :
( ( member_a @ A5 @ A2 )
=> ( ( F2 @ A5 )
= ( G2 @ A5 ) ) )
=> ( ( inj_on_a_complex @ F2 @ A2 )
= ( inj_on_a_complex @ G2 @ A2 ) ) ) ).
% inj_on_cong
thf(fact_163_inj__on__cong,axiom,
! [A2: set_complex,F2: complex > a,G2: complex > a] :
( ! [A5: complex] :
( ( member_complex @ A5 @ A2 )
=> ( ( F2 @ A5 )
= ( G2 @ A5 ) ) )
=> ( ( inj_on_complex_a @ F2 @ A2 )
= ( inj_on_complex_a @ G2 @ A2 ) ) ) ).
% inj_on_cong
thf(fact_164_inj__on__eq__iff,axiom,
! [F2: a > complex,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_complex @ F2 @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_165_inj__on__eq__iff,axiom,
! [F2: complex > a,A2: set_complex,X2: complex,Y: complex] :
( ( inj_on_complex_a @ F2 @ A2 )
=> ( ( member_complex @ X2 @ A2 )
=> ( ( member_complex @ Y @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_166_inj__on__contraD,axiom,
! [F2: a > complex,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_complex @ F2 @ A2 )
=> ( ( X2 != Y )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( F2 @ X2 )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_167_inj__on__contraD,axiom,
! [F2: complex > a,A2: set_complex,X2: complex,Y: complex] :
( ( inj_on_complex_a @ F2 @ A2 )
=> ( ( X2 != Y )
=> ( ( member_complex @ X2 @ A2 )
=> ( ( member_complex @ Y @ A2 )
=> ( ( F2 @ X2 )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_168_inj__on__inverseI,axiom,
! [A2: set_a,G2: complex > a,F2: a > complex] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( ( G2 @ ( F2 @ X4 ) )
= X4 ) )
=> ( inj_on_a_complex @ F2 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_169_inj__on__inverseI,axiom,
! [A2: set_complex,G2: a > complex,F2: complex > a] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A2 )
=> ( ( G2 @ ( F2 @ X4 ) )
= X4 ) )
=> ( inj_on_complex_a @ F2 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_170_inj__compose,axiom,
! [F2: real > a,G2: real > real] :
( ( inj_on_real_a @ F2 @ top_top_set_real )
=> ( ( inj_on_real_real @ G2 @ top_top_set_real )
=> ( inj_on_real_a @ ( comp_real_a_real @ F2 @ G2 ) @ top_top_set_real ) ) ) ).
% inj_compose
thf(fact_171_inj__compose,axiom,
! [F2: a > a,G2: real > a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( inj_on_real_a @ G2 @ top_top_set_real )
=> ( inj_on_real_a @ ( comp_a_a_real @ F2 @ G2 ) @ top_top_set_real ) ) ) ).
% inj_compose
thf(fact_172_inj__compose,axiom,
! [F2: a > complex,G2: real > a] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( ( inj_on_real_a @ G2 @ top_top_set_real )
=> ( inj_on_real_complex @ ( comp_a_complex_real @ F2 @ G2 ) @ top_top_set_real ) ) ) ).
% inj_compose
thf(fact_173_inj__compose,axiom,
! [F2: a > a,G2: a > a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( inj_on_a_a @ G2 @ top_top_set_a )
=> ( inj_on_a_a @ ( comp_a_a_a @ F2 @ G2 ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_174_inj__compose,axiom,
! [F2: a > complex,G2: a > a] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( ( inj_on_a_a @ G2 @ top_top_set_a )
=> ( inj_on_a_complex @ ( comp_a_complex_a @ F2 @ G2 ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_175_inj__compose,axiom,
! [F2: a > a,G2: complex > a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( inj_on_complex_a @ G2 @ top_top_set_complex )
=> ( inj_on_complex_a @ ( comp_a_a_complex @ F2 @ G2 ) @ top_top_set_complex ) ) ) ).
% inj_compose
thf(fact_176_inj__compose,axiom,
! [F2: a > complex,G2: complex > a] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( ( inj_on_complex_a @ G2 @ top_top_set_complex )
=> ( inj_on94911183omplex @ ( comp_a1063143865omplex @ F2 @ G2 ) @ top_top_set_complex ) ) ) ).
% inj_compose
thf(fact_177_inj__compose,axiom,
! [F2: complex > complex,G2: a > complex] :
( ( inj_on94911183omplex @ F2 @ top_top_set_complex )
=> ( ( inj_on_a_complex @ G2 @ top_top_set_a )
=> ( inj_on_a_complex @ ( comp_c124850173plex_a @ F2 @ G2 ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_178_inj__compose,axiom,
! [F2: complex > a,G2: a > complex] :
( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ( ( inj_on_a_complex @ G2 @ top_top_set_a )
=> ( inj_on_a_a @ ( comp_complex_a_a @ F2 @ G2 ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_179_inj__compose,axiom,
! [F2: complex > a,G2: complex > complex] :
( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ( ( inj_on94911183omplex @ G2 @ top_top_set_complex )
=> ( inj_on_complex_a @ ( comp_c274302683omplex @ F2 @ G2 ) @ top_top_set_complex ) ) ) ).
% inj_compose
thf(fact_180_inj__on__imageI2,axiom,
! [F3: a > complex,F2: real > a,A2: set_real] :
( ( inj_on_real_complex @ ( comp_a_complex_real @ F3 @ F2 ) @ A2 )
=> ( inj_on_real_a @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_181_inj__on__imageI2,axiom,
! [F3: real > a,F2: real > real,A2: set_real] :
( ( inj_on_real_a @ ( comp_real_a_real @ F3 @ F2 ) @ A2 )
=> ( inj_on_real_real @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_182_inj__on__imageI2,axiom,
! [F3: a > a,F2: real > a,A2: set_real] :
( ( inj_on_real_a @ ( comp_a_a_real @ F3 @ F2 ) @ A2 )
=> ( inj_on_real_a @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_183_inj__on__imageI2,axiom,
! [F3: a > a,F2: a > a,A2: set_a] :
( ( inj_on_a_a @ ( comp_a_a_a @ F3 @ F2 ) @ A2 )
=> ( inj_on_a_a @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_184_inj__on__imageI2,axiom,
! [F3: complex > a,F2: a > complex,A2: set_a] :
( ( inj_on_a_a @ ( comp_complex_a_a @ F3 @ F2 ) @ A2 )
=> ( inj_on_a_complex @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_185_inj__on__imageI2,axiom,
! [F3: a > complex,F2: complex > a,A2: set_complex] :
( ( inj_on94911183omplex @ ( comp_a1063143865omplex @ F3 @ F2 ) @ A2 )
=> ( inj_on_complex_a @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_186_inj__on__imageI2,axiom,
! [F3: a > complex,F2: a > a,A2: set_a] :
( ( inj_on_a_complex @ ( comp_a_complex_a @ F3 @ F2 ) @ A2 )
=> ( inj_on_a_a @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_187_inj__on__imageI2,axiom,
! [F3: complex > complex,F2: a > complex,A2: set_a] :
( ( inj_on_a_complex @ ( comp_c124850173plex_a @ F3 @ F2 ) @ A2 )
=> ( inj_on_a_complex @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_188_inj__on__imageI2,axiom,
! [F3: a > a,F2: complex > a,A2: set_complex] :
( ( inj_on_complex_a @ ( comp_a_a_complex @ F3 @ F2 ) @ A2 )
=> ( inj_on_complex_a @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_189_simple__path__linear__image__eq,axiom,
! [F2: real > a,G2: real > real] :
( ( real_V779700657real_a @ F2 )
=> ( ( inj_on_real_a @ F2 @ top_top_set_real )
=> ( ( path_simple_path_a @ ( comp_real_a_real @ F2 @ G2 ) )
= ( path_s1005760220h_real @ G2 ) ) ) ) ).
% simple_path_linear_image_eq
thf(fact_190_simple__path__linear__image__eq,axiom,
! [F2: a > a,G2: real > a] :
( ( real_V202220639ar_a_a @ F2 )
=> ( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( path_simple_path_a @ ( comp_a_a_real @ F2 @ G2 ) )
= ( path_simple_path_a @ G2 ) ) ) ) ).
% simple_path_linear_image_eq
thf(fact_191_simple__path__linear__image__eq,axiom,
! [F2: complex > a,G2: real > complex] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ( ( path_simple_path_a @ ( comp_complex_a_real @ F2 @ G2 ) )
= ( path_s36253918omplex @ G2 ) ) ) ) ).
% simple_path_linear_image_eq
thf(fact_192_simple__path__linear__image__eq,axiom,
! [F2: a > complex,G2: real > a] :
( ( real_V1477106445omplex @ F2 )
=> ( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( ( path_s36253918omplex @ ( comp_a_complex_real @ F2 @ G2 ) )
= ( path_simple_path_a @ G2 ) ) ) ) ).
% simple_path_linear_image_eq
thf(fact_193_simple__path__linear__image__eq,axiom,
! [F2: complex > complex,G2: real > complex] :
( ( real_V670066493omplex @ F2 )
=> ( ( inj_on94911183omplex @ F2 @ top_top_set_complex )
=> ( ( path_s36253918omplex @ ( comp_c595887981x_real @ F2 @ G2 ) )
= ( path_s36253918omplex @ G2 ) ) ) ) ).
% simple_path_linear_image_eq
thf(fact_194_fun_Oinj__map,axiom,
! [F2: real > a] :
( ( inj_on_real_a @ F2 @ top_top_set_real )
=> ( inj_on958237983real_a @ ( comp_real_a_real @ F2 ) @ top_to1446257885l_real ) ) ).
% fun.inj_map
thf(fact_195_fun_Oinj__map,axiom,
! [F2: a > a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( inj_on1576005937plex_a @ ( comp_a_a_complex @ F2 ) @ top_to525076535plex_a ) ) ).
% fun.inj_map
thf(fact_196_fun_Oinj__map,axiom,
! [F2: a > a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( inj_on_real_a_real_a @ ( comp_a_a_real @ F2 ) @ top_top_set_real_a ) ) ).
% fun.inj_map
thf(fact_197_fun_Oinj__map,axiom,
! [F2: a > a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( inj_on_a_a_a_a @ ( comp_a_a_a @ F2 ) @ top_top_set_a_a ) ) ).
% fun.inj_map
thf(fact_198_fun_Oinj__map,axiom,
! [F2: a > complex] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( inj_on319905617omplex @ ( comp_a_complex_real @ F2 ) @ top_top_set_real_a ) ) ).
% fun.inj_map
thf(fact_199_fun_Oinj__map,axiom,
! [F2: a > complex] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( inj_on893405649omplex @ ( comp_a1063143865omplex @ F2 ) @ top_to525076535plex_a ) ) ).
% fun.inj_map
thf(fact_200_fun_Oinj__map,axiom,
! [F2: a > complex] :
( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( inj_on_a_a_a_complex @ ( comp_a_complex_a @ F2 ) @ top_top_set_a_a ) ) ).
% fun.inj_map
thf(fact_201_fun_Oinj__map,axiom,
! [F2: complex > a] :
( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ( inj_on_a_complex_a_a @ ( comp_complex_a_a @ F2 ) @ top_to2109114701omplex ) ) ).
% fun.inj_map
thf(fact_202_real__of__bounded__linear,axiom,
real_V762982918plex_a @ poinca837721858l_of_a ).
% real_of_bounded_linear
thf(fact_203_real__of__linear,axiom,
real_V1327653935plex_a @ poinca837721858l_of_a ).
% real_of_linear
thf(fact_204_iso__tuple__UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_205_iso__tuple__UNIV__I,axiom,
! [X2: complex] : ( member_complex @ X2 @ top_top_set_complex ) ).
% iso_tuple_UNIV_I
thf(fact_206_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_207_UNIV__I,axiom,
! [X2: complex] : ( member_complex @ X2 @ top_top_set_complex ) ).
% UNIV_I
thf(fact_208_bounded__linear_Ointro,axiom,
! [F2: a > complex] :
( ( real_V1477106445omplex @ F2 )
=> ( ( real_V451440129omplex @ F2 )
=> ( real_V912435428omplex @ F2 ) ) ) ).
% bounded_linear.intro
thf(fact_209_bounded__linear_Ointro,axiom,
! [F2: complex > a] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( real_V301987619plex_a @ F2 )
=> ( real_V762982918plex_a @ F2 ) ) ) ).
% bounded_linear.intro
thf(fact_210_bounded__linear__def,axiom,
( real_V912435428omplex
= ( ^ [F: a > complex] :
( ( real_V1477106445omplex @ F )
& ( real_V451440129omplex @ F ) ) ) ) ).
% bounded_linear_def
thf(fact_211_bounded__linear__def,axiom,
( real_V762982918plex_a
= ( ^ [F: complex > a] :
( ( real_V1327653935plex_a @ F )
& ( real_V301987619plex_a @ F ) ) ) ) ).
% bounded_linear_def
thf(fact_212_arc__linear__image__eq,axiom,
! [F2: real > a,G2: real > real] :
( ( real_V779700657real_a @ F2 )
=> ( ( inj_on_real_a @ F2 @ top_top_set_real )
=> ( ( path_arc_a @ ( comp_real_a_real @ F2 @ G2 ) )
= ( path_arc_real @ G2 ) ) ) ) ).
% arc_linear_image_eq
thf(fact_213_arc__linear__image__eq,axiom,
! [F2: a > a,G2: real > a] :
( ( real_V202220639ar_a_a @ F2 )
=> ( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( path_arc_a @ ( comp_a_a_real @ F2 @ G2 ) )
= ( path_arc_a @ G2 ) ) ) ) ).
% arc_linear_image_eq
thf(fact_214_arc__linear__image__eq,axiom,
! [F2: a > complex,G2: real > a] :
( ( real_V1477106445omplex @ F2 )
=> ( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( ( path_arc_complex @ ( comp_a_complex_real @ F2 @ G2 ) )
= ( path_arc_a @ G2 ) ) ) ) ).
% arc_linear_image_eq
thf(fact_215_arc__linear__image__eq,axiom,
! [F2: complex > a,G2: real > complex] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ( ( path_arc_a @ ( comp_complex_a_real @ F2 @ G2 ) )
= ( path_arc_complex @ G2 ) ) ) ) ).
% arc_linear_image_eq
thf(fact_216_real__of__inj,axiom,
inj_on_complex_a @ poinca837721858l_of_a @ top_top_set_complex ).
% real_of_inj
thf(fact_217_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_218_top__set__def,axiom,
( top_top_set_complex
= ( collect_complex @ top_top_complex_o ) ) ).
% top_set_def
thf(fact_219_arc__imp__simple__path,axiom,
! [G2: real > a] :
( ( path_arc_a @ G2 )
=> ( path_simple_path_a @ G2 ) ) ).
% arc_imp_simple_path
thf(fact_220_arc__imp__simple__path,axiom,
! [G2: real > complex] :
( ( path_arc_complex @ G2 )
=> ( path_s36253918omplex @ G2 ) ) ).
% arc_imp_simple_path
thf(fact_221_arc__distinct__ends,axiom,
! [G2: real > complex] :
( ( path_arc_complex @ G2 )
=> ( ( path_p769714271omplex @ G2 )
!= ( path_p797330068omplex @ G2 ) ) ) ).
% arc_distinct_ends
thf(fact_222_arc__distinct__ends,axiom,
! [G2: real > a] :
( ( path_arc_a @ G2 )
=> ( ( path_pathfinish_a @ G2 )
!= ( path_pathstart_a @ G2 ) ) ) ).
% arc_distinct_ends
thf(fact_223_bounded__linear_Oaxioms_I2_J,axiom,
! [F2: a > complex] :
( ( real_V912435428omplex @ F2 )
=> ( real_V451440129omplex @ F2 ) ) ).
% bounded_linear.axioms(2)
thf(fact_224_bounded__linear_Oaxioms_I2_J,axiom,
! [F2: complex > a] :
( ( real_V762982918plex_a @ F2 )
=> ( real_V301987619plex_a @ F2 ) ) ).
% bounded_linear.axioms(2)
thf(fact_225_UNIV__eq__I,axiom,
! [A2: set_a] :
( ! [X4: a] : ( member_a @ X4 @ A2 )
=> ( top_top_set_a = A2 ) ) ).
% UNIV_eq_I
thf(fact_226_UNIV__eq__I,axiom,
! [A2: set_complex] :
( ! [X4: complex] : ( member_complex @ X4 @ A2 )
=> ( top_top_set_complex = A2 ) ) ).
% UNIV_eq_I
thf(fact_227_UNIV__witness,axiom,
? [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_228_UNIV__witness,axiom,
? [X4: complex] : ( member_complex @ X4 @ top_top_set_complex ) ).
% UNIV_witness
thf(fact_229_simple__path__imp__arc,axiom,
! [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_230_simple__path__imp__arc,axiom,
! [G2: real > complex] :
( ( path_s36253918omplex @ G2 )
=> ( ( ( path_p769714271omplex @ G2 )
!= ( path_p797330068omplex @ G2 ) )
=> ( path_arc_complex @ G2 ) ) ) ).
% simple_path_imp_arc
thf(fact_231_simple__path__eq__arc,axiom,
! [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_232_simple__path__eq__arc,axiom,
! [G2: real > complex] :
( ( ( path_p769714271omplex @ G2 )
!= ( path_p797330068omplex @ G2 ) )
=> ( ( path_s36253918omplex @ G2 )
= ( path_arc_complex @ G2 ) ) ) ).
% simple_path_eq_arc
thf(fact_233_simple__path__cases,axiom,
! [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_234_simple__path__cases,axiom,
! [G2: real > complex] :
( ( path_s36253918omplex @ G2 )
=> ( ( path_arc_complex @ G2 )
| ( ( path_p769714271omplex @ G2 )
= ( path_p797330068omplex @ G2 ) ) ) ) ).
% simple_path_cases
thf(fact_235_arc__simple__path,axiom,
( path_arc_a
= ( ^ [G: real > a] :
( ( path_simple_path_a @ G )
& ( ( path_pathfinish_a @ G )
!= ( path_pathstart_a @ G ) ) ) ) ) ).
% arc_simple_path
thf(fact_236_arc__simple__path,axiom,
( path_arc_complex
= ( ^ [G: real > complex] :
( ( path_s36253918omplex @ G )
& ( ( path_p769714271omplex @ G )
!= ( path_p797330068omplex @ G ) ) ) ) ) ).
% arc_simple_path
thf(fact_237_fun_Omap__comp,axiom,
! [G2: complex > complex,F2: a > complex,V: real > a] :
( ( comp_c595887981x_real @ G2 @ ( comp_a_complex_real @ F2 @ V ) )
= ( comp_a_complex_real @ ( comp_c124850173plex_a @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_238_fun_Omap__comp,axiom,
! [G2: complex > a,F2: a > complex,V: real > a] :
( ( comp_complex_a_real @ G2 @ ( comp_a_complex_real @ F2 @ V ) )
= ( comp_a_a_real @ ( comp_complex_a_a @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_239_fun_Omap__comp,axiom,
! [G2: complex > complex,F2: a > complex,V: complex > a] :
( ( comp_c130555887omplex @ G2 @ ( comp_a1063143865omplex @ F2 @ V ) )
= ( comp_a1063143865omplex @ ( comp_c124850173plex_a @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_240_fun_Omap__comp,axiom,
! [G2: complex > a,F2: a > complex,V: complex > a] :
( ( comp_c274302683omplex @ G2 @ ( comp_a1063143865omplex @ F2 @ V ) )
= ( comp_a_a_complex @ ( comp_complex_a_a @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_241_fun_Omap__comp,axiom,
! [G2: complex > complex,F2: a > complex,V: a > a] :
( ( comp_c124850173plex_a @ G2 @ ( comp_a_complex_a @ F2 @ V ) )
= ( comp_a_complex_a @ ( comp_c124850173plex_a @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_242_fun_Omap__comp,axiom,
! [G2: a > complex,F2: complex > a,V: real > complex] :
( ( comp_a_complex_real @ G2 @ ( comp_complex_a_real @ F2 @ V ) )
= ( comp_c595887981x_real @ ( comp_a1063143865omplex @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_243_fun_Omap__comp,axiom,
! [G2: a > complex,F2: real > a,V: real > real] :
( ( comp_a_complex_real @ G2 @ ( comp_real_a_real @ F2 @ V ) )
= ( comp_r701421291x_real @ ( comp_a_complex_real @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_244_fun_Omap__comp,axiom,
! [G2: a > complex,F2: a > a,V: real > a] :
( ( comp_a_complex_real @ G2 @ ( comp_a_a_real @ F2 @ V ) )
= ( comp_a_complex_real @ ( comp_a_complex_a @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_245_fun_Omap__comp,axiom,
! [G2: a > complex,F2: real > a,V: complex > real] :
( ( comp_a1063143865omplex @ G2 @ ( comp_real_a_complex @ F2 @ V ) )
= ( comp_r667767405omplex @ ( comp_a_complex_real @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_246_fun_Omap__comp,axiom,
! [G2: a > complex,F2: complex > a,V: complex > complex] :
( ( comp_a1063143865omplex @ G2 @ ( comp_c274302683omplex @ F2 @ V ) )
= ( comp_c130555887omplex @ ( comp_a1063143865omplex @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_247_complex__of__real__of,axiom,
( ( comp_a1063143865omplex @ poinca1910941596x_of_a @ poinca837721858l_of_a )
= id_complex ) ).
% complex_of_real_of
thf(fact_248_real__of__complex__of,axiom,
( ( comp_complex_a_a @ poinca837721858l_of_a @ poinca1910941596x_of_a )
= id_a ) ).
% real_of_complex_of
thf(fact_249_real__of__bij,axiom,
bij_betw_complex_a @ poinca837721858l_of_a @ top_top_set_complex @ top_top_set_a ).
% real_of_bij
thf(fact_250_complex__of__bij,axiom,
bij_betw_a_complex @ poinca1910941596x_of_a @ top_top_set_a @ top_top_set_complex ).
% complex_of_bij
thf(fact_251_linear__injective__left__inverse,axiom,
! [F2: a > a] :
( ( real_V202220639ar_a_a @ F2 )
=> ( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ? [G4: a > a] :
( ( real_V202220639ar_a_a @ G4 )
& ( ( comp_a_a_a @ G4 @ F2 )
= id_a ) ) ) ) ).
% linear_injective_left_inverse
thf(fact_252_linear__injective__left__inverse,axiom,
! [F2: a > complex] :
( ( real_V1477106445omplex @ F2 )
=> ( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ? [G4: complex > a] :
( ( real_V1327653935plex_a @ G4 )
& ( ( comp_complex_a_a @ G4 @ F2 )
= id_a ) ) ) ) ).
% linear_injective_left_inverse
thf(fact_253_linear__injective__left__inverse,axiom,
! [F2: complex > a] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ? [G4: a > complex] :
( ( real_V1477106445omplex @ G4 )
& ( ( comp_a1063143865omplex @ G4 @ F2 )
= id_complex ) ) ) ) ).
% linear_injective_left_inverse
thf(fact_254_path__linear__image__eq,axiom,
! [F2: real > a,G2: real > real] :
( ( real_V779700657real_a @ F2 )
=> ( ( inj_on_real_a @ F2 @ top_top_set_real )
=> ( ( path_path_a @ ( comp_real_a_real @ F2 @ G2 ) )
= ( path_path_real @ G2 ) ) ) ) ).
% path_linear_image_eq
thf(fact_255_path__linear__image__eq,axiom,
! [F2: a > a,G2: real > a] :
( ( real_V202220639ar_a_a @ F2 )
=> ( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( path_path_a @ ( comp_a_a_real @ F2 @ G2 ) )
= ( path_path_a @ G2 ) ) ) ) ).
% path_linear_image_eq
thf(fact_256_path__linear__image__eq,axiom,
! [F2: complex > complex,G2: real > complex] :
( ( real_V670066493omplex @ F2 )
=> ( ( inj_on94911183omplex @ F2 @ top_top_set_complex )
=> ( ( path_path_complex @ ( comp_c595887981x_real @ F2 @ G2 ) )
= ( path_path_complex @ G2 ) ) ) ) ).
% path_linear_image_eq
thf(fact_257_path__linear__image__eq,axiom,
! [F2: a > complex,G2: real > a] :
( ( real_V1477106445omplex @ F2 )
=> ( ( inj_on_a_complex @ F2 @ top_top_set_a )
=> ( ( path_path_complex @ ( comp_a_complex_real @ F2 @ G2 ) )
= ( path_path_a @ G2 ) ) ) ) ).
% path_linear_image_eq
thf(fact_258_path__linear__image__eq,axiom,
! [F2: complex > a,G2: real > complex] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( inj_on_complex_a @ F2 @ top_top_set_complex )
=> ( ( path_path_a @ ( comp_complex_a_real @ F2 @ G2 ) )
= ( path_path_complex @ G2 ) ) ) ) ).
% path_linear_image_eq
thf(fact_259_linear__inj__iff__eq__0,axiom,
! [F2: a > complex] :
( ( real_V1477106445omplex @ F2 )
=> ( ( inj_on_a_complex @ F2 @ top_top_set_a )
= ( ! [X: a] :
( ( ( F2 @ X )
= zero_zero_complex )
=> ( X = zero_zero_a ) ) ) ) ) ).
% linear_inj_iff_eq_0
thf(fact_260_linear__inj__iff__eq__0,axiom,
! [F2: complex > a] :
( ( real_V1327653935plex_a @ F2 )
=> ( ( inj_on_complex_a @ F2 @ top_top_set_complex )
= ( ! [X: complex] :
( ( ( F2 @ X )
= zero_zero_a )
=> ( X = zero_zero_complex ) ) ) ) ) ).
% linear_inj_iff_eq_0
thf(fact_261_type__copy__map__cong0,axiom,
! [M: real > a,G2: real > real,X2: real,N: a > a,H: real > a,F2: a > complex] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_r701421291x_real @ ( comp_a_complex_real @ F2 @ M ) @ G2 @ X2 )
= ( comp_a_complex_real @ ( comp_a_complex_a @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_262_type__copy__map__cong0,axiom,
! [M: real > a,G2: complex > real,X2: complex,N: a > a,H: complex > a,F2: a > complex] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_r667767405omplex @ ( comp_a_complex_real @ F2 @ M ) @ G2 @ X2 )
= ( comp_a1063143865omplex @ ( comp_a_complex_a @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_263_type__copy__map__cong0,axiom,
! [M: real > a,G2: a > real,X2: a,N: a > a,H: a > a,F2: a > complex] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_real_complex_a @ ( comp_a_complex_real @ F2 @ M ) @ G2 @ X2 )
= ( comp_a_complex_a @ ( comp_a_complex_a @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_264_type__copy__map__cong0,axiom,
! [M: complex > a,G2: real > complex,X2: real,N: a > a,H: real > a,F2: a > complex] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_c595887981x_real @ ( comp_a1063143865omplex @ F2 @ M ) @ G2 @ X2 )
= ( comp_a_complex_real @ ( comp_a_complex_a @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_265_type__copy__map__cong0,axiom,
! [M: complex > a,G2: complex > complex,X2: complex,N: a > a,H: complex > a,F2: a > complex] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_c130555887omplex @ ( comp_a1063143865omplex @ F2 @ M ) @ G2 @ X2 )
= ( comp_a1063143865omplex @ ( comp_a_complex_a @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_266_type__copy__map__cong0,axiom,
! [M: complex > a,G2: a > complex,X2: a,N: a > a,H: a > a,F2: a > complex] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_c124850173plex_a @ ( comp_a1063143865omplex @ F2 @ M ) @ G2 @ X2 )
= ( comp_a_complex_a @ ( comp_a_complex_a @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_267_type__copy__map__cong0,axiom,
! [M: real > real,G2: a > real,X2: a,N: complex > real,H: a > complex,F2: real > a] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_real_a_a @ ( comp_real_a_real @ F2 @ M ) @ G2 @ X2 )
= ( comp_complex_a_a @ ( comp_real_a_complex @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_268_type__copy__map__cong0,axiom,
! [M: real > real,G2: complex > real,X2: complex,N: a > real,H: complex > a,F2: real > a] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_real_a_complex @ ( comp_real_a_real @ F2 @ M ) @ G2 @ X2 )
= ( comp_a_a_complex @ ( comp_real_a_a @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_269_type__copy__map__cong0,axiom,
! [M: real > real,G2: a > real,X2: a,N: a > real,H: a > a,F2: real > a] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_real_a_a @ ( comp_real_a_real @ F2 @ M ) @ G2 @ X2 )
= ( comp_a_a_a @ ( comp_real_a_a @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_270_type__copy__map__cong0,axiom,
! [M: complex > a,G2: real > complex,X2: real,N: real > a,H: real > real,F2: a > a] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_complex_a_real @ ( comp_a_a_complex @ F2 @ M ) @ G2 @ X2 )
= ( comp_real_a_real @ ( comp_a_a_real @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_271_id__apply,axiom,
( id_complex
= ( ^ [X: complex] : X ) ) ).
% id_apply
thf(fact_272_id__apply,axiom,
( id_a
= ( ^ [X: a] : X ) ) ).
% id_apply
thf(fact_273_path__rectpath,axiom,
! [A: complex,B: complex] : ( path_path_complex @ ( path_rectpath @ A @ B ) ) ).
% path_rectpath
thf(fact_274_comp__id,axiom,
! [F2: real > a] :
( ( comp_real_a_real @ F2 @ id_real )
= F2 ) ).
% comp_id
thf(fact_275_comp__id,axiom,
! [F2: a > complex] :
( ( comp_a_complex_a @ F2 @ id_a )
= F2 ) ).
% comp_id
thf(fact_276_comp__id,axiom,
! [F2: a > a] :
( ( comp_a_a_a @ F2 @ id_a )
= F2 ) ).
% comp_id
thf(fact_277_id__comp,axiom,
! [G2: complex > a] :
( ( comp_a_a_complex @ id_a @ G2 )
= G2 ) ).
% id_comp
thf(fact_278_id__comp,axiom,
! [G2: real > a] :
( ( comp_a_a_real @ id_a @ G2 )
= G2 ) ).
% id_comp
thf(fact_279_id__comp,axiom,
! [G2: a > a] :
( ( comp_a_a_a @ id_a @ G2 )
= G2 ) ).
% id_comp
thf(fact_280_fun_Omap__id,axiom,
! [T: complex > a] :
( ( comp_a_a_complex @ id_a @ T )
= T ) ).
% fun.map_id
thf(fact_281_fun_Omap__id,axiom,
! [T: real > a] :
( ( comp_a_a_real @ id_a @ T )
= T ) ).
% fun.map_id
thf(fact_282_fun_Omap__id,axiom,
! [T: a > a] :
( ( comp_a_a_a @ id_a @ T )
= T ) ).
% fun.map_id
thf(fact_283_bij__betw__id,axiom,
! [A2: set_complex] : ( bij_be209634132omplex @ id_complex @ A2 @ A2 ) ).
% bij_betw_id
thf(fact_284_bij__betw__id,axiom,
! [A2: set_a] : ( bij_betw_a_a @ id_a @ A2 @ A2 ) ).
% bij_betw_id
thf(fact_285_bij__id,axiom,
bij_betw_a_a @ id_a @ top_top_set_a @ top_top_set_a ).
% bij_id
thf(fact_286_bij__id,axiom,
bij_be209634132omplex @ id_complex @ top_top_set_complex @ top_top_set_complex ).
% bij_id
thf(fact_287_DEADID_Oin__rel,axiom,
( ( ^ [Y4: a,Z: a] : Y4 = Z )
= ( ^ [A6: a,B2: a] :
? [Z2: a] :
( ( member_a @ Z2 @ top_top_set_a )
& ( ( id_a @ Z2 )
= A6 )
& ( ( id_a @ Z2 )
= B2 ) ) ) ) ).
% DEADID.in_rel
thf(fact_288_DEADID_Oin__rel,axiom,
( ( ^ [Y4: complex,Z: complex] : Y4 = Z )
= ( ^ [A6: complex,B2: complex] :
? [Z2: complex] :
( ( member_complex @ Z2 @ top_top_set_complex )
& ( ( id_complex @ Z2 )
= A6 )
& ( ( id_complex @ Z2 )
= B2 ) ) ) ) ).
% DEADID.in_rel
thf(fact_289_fun_Omap__id0,axiom,
( ( comp_a_a_complex @ id_a )
= id_complex_a ) ).
% fun.map_id0
thf(fact_290_fun_Omap__id0,axiom,
( ( comp_a_a_real @ id_a )
= id_real_a ) ).
% fun.map_id0
thf(fact_291_fun_Omap__id0,axiom,
( ( comp_a_a_a @ id_a )
= id_a_a ) ).
% fun.map_id0
thf(fact_292_id__def,axiom,
( id_complex
= ( ^ [X: complex] : X ) ) ).
% id_def
thf(fact_293_id__def,axiom,
( id_a
= ( ^ [X: a] : X ) ) ).
% id_def
thf(fact_294_bij__betwE,axiom,
! [F2: complex > a,A2: set_complex,B3: set_a] :
( ( bij_betw_complex_a @ F2 @ A2 @ B3 )
=> ! [X5: complex] :
( ( member_complex @ X5 @ A2 )
=> ( member_a @ ( F2 @ X5 ) @ B3 ) ) ) ).
% bij_betwE
thf(fact_295_bij__betwE,axiom,
! [F2: a > complex,A2: set_a,B3: set_complex] :
( ( bij_betw_a_complex @ F2 @ A2 @ B3 )
=> ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( member_complex @ ( F2 @ X5 ) @ B3 ) ) ) ).
% bij_betwE
thf(fact_296_eq__id__iff,axiom,
! [F2: complex > complex] :
( ( ! [X: complex] :
( ( F2 @ X )
= X ) )
= ( F2 = id_complex ) ) ).
% eq_id_iff
thf(fact_297_eq__id__iff,axiom,
! [F2: a > a] :
( ( ! [X: a] :
( ( F2 @ X )
= X ) )
= ( F2 = id_a ) ) ).
% eq_id_iff
thf(fact_298_bij__betw__inv,axiom,
! [F2: complex > a,A2: set_complex,B3: set_a] :
( ( bij_betw_complex_a @ F2 @ A2 @ B3 )
=> ? [G4: a > complex] : ( bij_betw_a_complex @ G4 @ B3 @ A2 ) ) ).
% bij_betw_inv
thf(fact_299_bij__betw__inv,axiom,
! [F2: a > complex,A2: set_a,B3: set_complex] :
( ( bij_betw_a_complex @ F2 @ A2 @ B3 )
=> ? [G4: complex > a] : ( bij_betw_complex_a @ G4 @ B3 @ A2 ) ) ).
% bij_betw_inv
thf(fact_300_bij__betw__cong,axiom,
! [A2: set_complex,F2: complex > a,G2: complex > a,A7: set_a] :
( ! [A5: complex] :
( ( member_complex @ A5 @ A2 )
=> ( ( F2 @ A5 )
= ( G2 @ A5 ) ) )
=> ( ( bij_betw_complex_a @ F2 @ A2 @ A7 )
= ( bij_betw_complex_a @ G2 @ A2 @ A7 ) ) ) ).
% bij_betw_cong
thf(fact_301_bij__betw__cong,axiom,
! [A2: set_a,F2: a > complex,G2: a > complex,A7: set_complex] :
( ! [A5: a] :
( ( member_a @ A5 @ A2 )
=> ( ( F2 @ A5 )
= ( G2 @ A5 ) ) )
=> ( ( bij_betw_a_complex @ F2 @ A2 @ A7 )
= ( bij_betw_a_complex @ G2 @ A2 @ A7 ) ) ) ).
% bij_betw_cong
thf(fact_302_bij__betw__apply,axiom,
! [F2: complex > a,A2: set_complex,B3: set_a,A: complex] :
( ( bij_betw_complex_a @ F2 @ A2 @ B3 )
=> ( ( member_complex @ A @ A2 )
=> ( member_a @ ( F2 @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_303_bij__betw__apply,axiom,
! [F2: a > complex,A2: set_a,B3: set_complex,A: a] :
( ( bij_betw_a_complex @ F2 @ A2 @ B3 )
=> ( ( member_a @ A @ A2 )
=> ( member_complex @ ( F2 @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_304_bij__betw__id__iff,axiom,
! [A2: set_complex,B3: set_complex] :
( ( bij_be209634132omplex @ id_complex @ A2 @ B3 )
= ( A2 = B3 ) ) ).
% bij_betw_id_iff
thf(fact_305_bij__betw__id__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( bij_betw_a_a @ id_a @ A2 @ B3 )
= ( A2 = B3 ) ) ).
% bij_betw_id_iff
thf(fact_306_bij__betw__iff__bijections,axiom,
( bij_betw_complex_a
= ( ^ [F: complex > a,A4: set_complex,B4: set_a] :
? [G: a > complex] :
( ! [X: complex] :
( ( member_complex @ X @ A4 )
=> ( ( member_a @ ( F @ X ) @ B4 )
& ( ( G @ ( F @ X ) )
= X ) ) )
& ! [X: a] :
( ( member_a @ X @ B4 )
=> ( ( member_complex @ ( G @ X ) @ A4 )
& ( ( F @ ( G @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_307_bij__betw__iff__bijections,axiom,
( bij_betw_a_complex
= ( ^ [F: a > complex,A4: set_a,B4: set_complex] :
? [G: complex > a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ( member_complex @ ( F @ X ) @ B4 )
& ( ( G @ ( F @ X ) )
= X ) ) )
& ! [X: complex] :
( ( member_complex @ X @ B4 )
=> ( ( member_a @ ( G @ X ) @ A4 )
& ( ( F @ ( G @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_308_o__bij,axiom,
! [G2: a > a,F2: a > a] :
( ( ( comp_a_a_a @ G2 @ F2 )
= id_a )
=> ( ( ( comp_a_a_a @ F2 @ G2 )
= id_a )
=> ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a ) ) ) ).
% o_bij
thf(fact_309_o__bij,axiom,
! [G2: complex > a,F2: a > complex] :
( ( ( comp_complex_a_a @ G2 @ F2 )
= id_a )
=> ( ( ( comp_a1063143865omplex @ F2 @ G2 )
= id_complex )
=> ( bij_betw_a_complex @ F2 @ top_top_set_a @ top_top_set_complex ) ) ) ).
% o_bij
thf(fact_310_o__bij,axiom,
! [G2: a > complex,F2: complex > a] :
( ( ( comp_a1063143865omplex @ G2 @ F2 )
= id_complex )
=> ( ( ( comp_complex_a_a @ F2 @ G2 )
= id_a )
=> ( bij_betw_complex_a @ F2 @ top_top_set_complex @ top_top_set_a ) ) ) ).
% o_bij
thf(fact_311_o__bij,axiom,
! [G2: complex > complex,F2: complex > complex] :
( ( ( comp_c130555887omplex @ G2 @ F2 )
= id_complex )
=> ( ( ( comp_c130555887omplex @ F2 @ G2 )
= id_complex )
=> ( bij_be209634132omplex @ F2 @ top_top_set_complex @ top_top_set_complex ) ) ) ).
% o_bij
thf(fact_312_bij__pointE,axiom,
! [F2: a > a,Y: a] :
( ( bij_betw_a_a @ F2 @ top_top_set_a @ top_top_set_a )
=> ~ ! [X4: a] :
( ( Y
= ( F2 @ X4 ) )
=> ~ ! [X6: a] :
( ( Y
= ( F2 @ X6 ) )
=> ( X6 = X4 ) ) ) ) ).
% bij_pointE
thf(fact_313_bij__pointE,axiom,
! [F2: a > complex,Y: complex] :
( ( bij_betw_a_complex @ F2 @ top_top_set_a @ top_top_set_complex )
=> ~ ! [X4: a] :
( ( Y
= ( F2 @ X4 ) )
=> ~ ! [X6: a] :
( ( Y
= ( F2 @ X6 ) )
=> ( X6 = X4 ) ) ) ) ).
% bij_pointE
thf(fact_314_bij__pointE,axiom,
! [F2: complex > a,Y: a] :
( ( bij_betw_complex_a @ F2 @ top_top_set_complex @ top_top_set_a )
=> ~ ! [X4: complex] :
( ( Y
= ( F2 @ X4 ) )
=> ~ ! [X6: complex] :
( ( Y
= ( F2 @ X6 ) )
=> ( X6 = X4 ) ) ) ) ).
% bij_pointE
thf(fact_315_bij__pointE,axiom,
! [F2: complex > complex,Y: complex] :
( ( bij_be209634132omplex @ F2 @ top_top_set_complex @ top_top_set_complex )
=> ~ ! [X4: complex] :
( ( Y
= ( F2 @ X4 ) )
=> ~ ! [X6: complex] :
( ( Y
= ( F2 @ X6 ) )
=> ( X6 = X4 ) ) ) ) ).
% bij_pointE
thf(fact_316_bij__betw__trans,axiom,
! [F2: real > real,A2: set_real,B3: set_real,G2: real > a,C2: set_a] :
( ( bij_betw_real_real @ F2 @ A2 @ B3 )
=> ( ( bij_betw_real_a @ G2 @ B3 @ C2 )
=> ( bij_betw_real_a @ ( comp_real_a_real @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_317_bij__betw__trans,axiom,
! [F2: real > a,A2: set_real,B3: set_a,G2: a > a,C2: set_a] :
( ( bij_betw_real_a @ F2 @ A2 @ B3 )
=> ( ( bij_betw_a_a @ G2 @ B3 @ C2 )
=> ( bij_betw_real_a @ ( comp_a_a_real @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_318_bij__betw__trans,axiom,
! [F2: a > a,A2: set_a,B3: set_a,G2: a > a,C2: set_a] :
( ( bij_betw_a_a @ F2 @ A2 @ B3 )
=> ( ( bij_betw_a_a @ G2 @ B3 @ C2 )
=> ( bij_betw_a_a @ ( comp_a_a_a @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_319_bij__betw__trans,axiom,
! [F2: complex > complex,A2: set_complex,B3: set_complex,G2: complex > a,C2: set_a] :
( ( bij_be209634132omplex @ F2 @ A2 @ B3 )
=> ( ( bij_betw_complex_a @ G2 @ B3 @ C2 )
=> ( bij_betw_complex_a @ ( comp_c274302683omplex @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_320_bij__betw__trans,axiom,
! [F2: real > a,A2: set_real,B3: set_a,G2: a > complex,C2: set_complex] :
( ( bij_betw_real_a @ F2 @ A2 @ B3 )
=> ( ( bij_betw_a_complex @ G2 @ B3 @ C2 )
=> ( bij_be122140626omplex @ ( comp_a_complex_real @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_321_bij__betw__trans,axiom,
! [F2: a > a,A2: set_a,B3: set_a,G2: a > complex,C2: set_complex] :
( ( bij_betw_a_a @ F2 @ A2 @ B3 )
=> ( ( bij_betw_a_complex @ G2 @ B3 @ C2 )
=> ( bij_betw_a_complex @ ( comp_a_complex_a @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_322_bij__betw__trans,axiom,
! [F2: complex > a,A2: set_complex,B3: set_a,G2: a > a,C2: set_a] :
( ( bij_betw_complex_a @ F2 @ A2 @ B3 )
=> ( ( bij_betw_a_a @ G2 @ B3 @ C2 )
=> ( bij_betw_complex_a @ ( comp_a_a_complex @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_323_bij__betw__trans,axiom,
! [F2: complex > a,A2: set_complex,B3: set_a,G2: a > complex,C2: set_complex] :
( ( bij_betw_complex_a @ F2 @ A2 @ B3 )
=> ( ( bij_betw_a_complex @ G2 @ B3 @ C2 )
=> ( bij_be209634132omplex @ ( comp_a1063143865omplex @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_324_bij__betw__trans,axiom,
! [F2: a > complex,A2: set_a,B3: set_complex,G2: complex > complex,C2: set_complex] :
( ( bij_betw_a_complex @ F2 @ A2 @ B3 )
=> ( ( bij_be209634132omplex @ G2 @ B3 @ C2 )
=> ( bij_betw_a_complex @ ( comp_c124850173plex_a @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_325_bij__betw__trans,axiom,
! [F2: a > complex,A2: set_a,B3: set_complex,G2: complex > a,C2: set_a] :
( ( bij_betw_a_complex @ F2 @ A2 @ B3 )
=> ( ( bij_betw_complex_a @ G2 @ B3 @ C2 )
=> ( bij_betw_a_a @ ( comp_complex_a_a @ G2 @ F2 ) @ A2 @ C2 ) ) ) ).
% bij_betw_trans
thf(fact_326_bij__betw__comp__iff,axiom,
! [F2: real > real,A2: set_real,A7: set_real,F3: real > a,A8: set_a] :
( ( bij_betw_real_real @ F2 @ A2 @ A7 )
=> ( ( bij_betw_real_a @ F3 @ A7 @ A8 )
= ( bij_betw_real_a @ ( comp_real_a_real @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_327_bij__betw__comp__iff,axiom,
! [F2: real > a,A2: set_real,A7: set_a,F3: a > a,A8: set_a] :
( ( bij_betw_real_a @ F2 @ A2 @ A7 )
=> ( ( bij_betw_a_a @ F3 @ A7 @ A8 )
= ( bij_betw_real_a @ ( comp_a_a_real @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_328_bij__betw__comp__iff,axiom,
! [F2: a > a,A2: set_a,A7: set_a,F3: a > a,A8: set_a] :
( ( bij_betw_a_a @ F2 @ A2 @ A7 )
=> ( ( bij_betw_a_a @ F3 @ A7 @ A8 )
= ( bij_betw_a_a @ ( comp_a_a_a @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_329_bij__betw__comp__iff,axiom,
! [F2: complex > complex,A2: set_complex,A7: set_complex,F3: complex > a,A8: set_a] :
( ( bij_be209634132omplex @ F2 @ A2 @ A7 )
=> ( ( bij_betw_complex_a @ F3 @ A7 @ A8 )
= ( bij_betw_complex_a @ ( comp_c274302683omplex @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_330_bij__betw__comp__iff,axiom,
! [F2: real > a,A2: set_real,A7: set_a,F3: a > complex,A8: set_complex] :
( ( bij_betw_real_a @ F2 @ A2 @ A7 )
=> ( ( bij_betw_a_complex @ F3 @ A7 @ A8 )
= ( bij_be122140626omplex @ ( comp_a_complex_real @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_331_bij__betw__comp__iff,axiom,
! [F2: a > a,A2: set_a,A7: set_a,F3: a > complex,A8: set_complex] :
( ( bij_betw_a_a @ F2 @ A2 @ A7 )
=> ( ( bij_betw_a_complex @ F3 @ A7 @ A8 )
= ( bij_betw_a_complex @ ( comp_a_complex_a @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_332_bij__betw__comp__iff,axiom,
! [F2: complex > a,A2: set_complex,A7: set_a,F3: a > a,A8: set_a] :
( ( bij_betw_complex_a @ F2 @ A2 @ A7 )
=> ( ( bij_betw_a_a @ F3 @ A7 @ A8 )
= ( bij_betw_complex_a @ ( comp_a_a_complex @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_333_bij__betw__comp__iff,axiom,
! [F2: complex > a,A2: set_complex,A7: set_a,F3: a > complex,A8: set_complex] :
( ( bij_betw_complex_a @ F2 @ A2 @ A7 )
=> ( ( bij_betw_a_complex @ F3 @ A7 @ A8 )
= ( bij_be209634132omplex @ ( comp_a1063143865omplex @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_334_bij__betw__comp__iff,axiom,
! [F2: a > complex,A2: set_a,A7: set_complex,F3: complex > complex,A8: set_complex] :
( ( bij_betw_a_complex @ F2 @ A2 @ A7 )
=> ( ( bij_be209634132omplex @ F3 @ A7 @ A8 )
= ( bij_betw_a_complex @ ( comp_c124850173plex_a @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_335_bij__betw__comp__iff,axiom,
! [F2: a > complex,A2: set_a,A7: set_complex,F3: complex > a,A8: set_a] :
( ( bij_betw_a_complex @ F2 @ A2 @ A7 )
=> ( ( bij_betw_complex_a @ F3 @ A7 @ A8 )
= ( bij_betw_a_a @ ( comp_complex_a_a @ F3 @ F2 ) @ A2 @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_336_bij__betw__imp__inj__on,axiom,
! [F2: complex > a,A2: set_complex,B3: set_a] :
( ( bij_betw_complex_a @ F2 @ A2 @ B3 )
=> ( inj_on_complex_a @ F2 @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_337_bij__betw__imp__inj__on,axiom,
! [F2: a > complex,A2: set_a,B3: set_complex] :
( ( bij_betw_a_complex @ F2 @ A2 @ B3 )
=> ( inj_on_a_complex @ F2 @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_338_pointfree__idE,axiom,
! [F2: a > complex,G2: complex > a,X2: complex] :
( ( ( comp_a1063143865omplex @ F2 @ G2 )
= id_complex )
=> ( ( F2 @ ( G2 @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_339_pointfree__idE,axiom,
! [F2: complex > a,G2: a > complex,X2: a] :
( ( ( comp_complex_a_a @ F2 @ G2 )
= id_a )
=> ( ( F2 @ ( G2 @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_340_pointfree__idE,axiom,
! [F2: a > a,G2: a > a,X2: a] :
( ( ( comp_a_a_a @ F2 @ G2 )
= id_a )
=> ( ( F2 @ ( G2 @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_341_comp__eq__id__dest,axiom,
! [A: a > complex,B: real > a,C: real > complex,V: real] :
( ( ( comp_a_complex_real @ A @ B )
= ( comp_c595887981x_real @ id_complex @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_342_comp__eq__id__dest,axiom,
! [A: a > complex,B: complex > a,C: complex > complex,V: complex] :
( ( ( comp_a1063143865omplex @ A @ B )
= ( comp_c130555887omplex @ id_complex @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_343_comp__eq__id__dest,axiom,
! [A: complex > a,B: a > complex,C: a > a,V: a] :
( ( ( comp_complex_a_a @ A @ B )
= ( comp_a_a_a @ id_a @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_344_comp__eq__id__dest,axiom,
! [A: real > a,B: real > real,C: real > a,V: real] :
( ( ( comp_real_a_real @ A @ B )
= ( comp_a_a_real @ id_a @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_345_comp__eq__id__dest,axiom,
! [A: a > complex,B: a > a,C: a > complex,V: a] :
( ( ( comp_a_complex_a @ A @ B )
= ( comp_c124850173plex_a @ id_complex @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_346_comp__eq__id__dest,axiom,
! [A: a > a,B: complex > a,C: complex > a,V: complex] :
( ( ( comp_a_a_complex @ A @ B )
= ( comp_a_a_complex @ id_a @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_347_comp__eq__id__dest,axiom,
! [A: a > a,B: real > a,C: real > a,V: real] :
( ( ( comp_a_a_real @ A @ B )
= ( comp_a_a_real @ id_a @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_348_comp__eq__id__dest,axiom,
! [A: a > a,B: a > a,C: a > a,V: a] :
( ( ( comp_a_a_a @ A @ B )
= ( comp_a_a_a @ id_a @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_349_arc__imp__path,axiom,
! [G2: real > complex] :
( ( path_arc_complex @ G2 )
=> ( path_path_complex @ G2 ) ) ).
% arc_imp_path
thf(fact_350_inj__on__id,axiom,
! [A2: set_complex] : ( inj_on94911183omplex @ id_complex @ A2 ) ).
% inj_on_id
thf(fact_351_inj__on__id,axiom,
! [A2: set_a] : ( inj_on_a_a @ id_a @ A2 ) ).
% inj_on_id
thf(fact_352_simple__path__imp__path,axiom,
! [G2: real > a] :
( ( path_simple_path_a @ G2 )
=> ( path_path_a @ G2 ) ) ).
% simple_path_imp_path
thf(fact_353_simple__path__imp__path,axiom,
! [G2: real > complex] :
( ( path_s36253918omplex @ G2 )
=> ( path_path_complex @ G2 ) ) ).
% simple_path_imp_path
% Conjectures (1)
thf(conj_0,conjecture,
( ( path_p769714271omplex @ ( comp_a_complex_real @ poinca1910941596x_of_a @ c ) )
= ( path_p797330068omplex @ ( comp_a_complex_real @ poinca1910941596x_of_a @ c ) ) ) ).
%------------------------------------------------------------------------------