%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP148^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:56 AM UTC 2026
% Result : Theorem 3.93s 2.11s
% Output : Refutation 3.93s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP148^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n006.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 08:27:45 EST 2026
% 0.14/0.32 % CPUTime :
% 0.14/0.32 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.43 ---- Original DTF file ---
% 0.17/0.43 thf(spec,logic,$$dhol).
% 0.17/0.43 %------------------------------------------------------------------------------
% 0.17/0.43 % File : ITP148^2 : TPTP v9.2.1. Released v7.5.0.
% 0.17/0.43 % Domain : Interactive Theorem Proving
% 0.17/0.43 % Problem : Sledgehammer Poincare_Bendixson problem prob_301__19573322_1
% 0.17/0.43 % Version : Especial.
% 0.17/0.43 % English :
% 0.17/0.43
% 0.17/0.43 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.17/0.43 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.17/0.43 % Source : [Des21]
% 0.17/0.43 % Names : Poincare_Bendixson/prob_301__19573322_1 [Des21]
% 0.17/0.43
% 0.17/0.43 % Status : Theorem
% 0.17/0.43 % Rating : 0.00 v7.5.0
% 0.17/0.43 % Syntax : Number of formulae : 330 ( 89 unt; 45 typ; 0 def)
% 0.17/0.43 % Number of atoms : 813 ( 349 equ; 0 cnn)
% 0.17/0.43 % Maximal formula atoms : 9 ( 2 avg)
% 0.17/0.43 % Number of connectives : 4129 ( 35 ~; 2 |; 64 &;3664 @)
% 0.17/0.43 % ( 0 <=>; 364 =>; 0 <=; 0 <~>)
% 0.17/0.43 % Maximal formula depth : 19 ( 8 avg)
% 0.17/0.43 % Number of types : 5 ( 4 usr)
% 0.17/0.43 % Number of type conns : 423 ( 423 >; 0 *; 0 +; 0 <<)
% 0.17/0.43 % Number of symbols : 42 ( 41 usr; 0 con; 1-8 aty)
% 0.17/0.43 % Number of variables : 1184 ( 32 ^;1083 !; 19 ?;1184 :)
% 0.17/0.43 % ( 50 !>; 0 ?*; 0 @-; 0 @+)
% 0.17/0.43 % SPC : TH1_THM_EQU_NAR
% 0.17/0.43
% 0.17/0.43 % Comments : This file was generated by Sledgehammer 2021-02-23 16:27:43.127
% 0.17/0.43 %------------------------------------------------------------------------------
% 0.17/0.43 % Could-be-implicit typings (6)
% 0.17/0.43 thf(ty_t_Bounded__Linear__Function_Oblinfun,type,
% 0.17/0.43 bounde2145540817linfun: $tType > $tType > $tType ).
% 0.17/0.43
% 0.17/0.43 thf(ty_t_Complex_Ocomplex,type,
% 0.17/0.43 complex: $tType ).
% 0.17/0.43
% 0.17/0.43 thf(ty_t_Real_Oreal,type,
% 0.17/0.43 real: $tType ).
% 0.17/0.43
% 0.17/0.43 thf(ty_t_Set_Oset,type,
% 0.17/0.43 set: $tType > $tType ).
% 0.17/0.43
% 0.17/0.43 thf(ty_t_Int_Oint,type,
% 0.17/0.43 int: $tType ).
% 0.17/0.43
% 0.17/0.43 thf(ty_tf_a,type,
% 0.17/0.43 a: $tType ).
% 0.17/0.43
% 0.17/0.43 % Explicit typings (39)
% 0.17/0.43 thf(sy_cl_HOL_Otype,type,
% 0.17/0.43 type:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_cl_Groups_Ozero,type,
% 0.17/0.43 zero:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_cl_Orderings_Otop,type,
% 0.17/0.43 top:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_cl_Inner__Product_Oreal__inner,type,
% 0.17/0.43 inner_real_inner:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
% 0.17/0.43 real_V1076094709vector:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
% 0.17/0.43 euclid925273238_space:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 0.17/0.43 topolo503727757_space:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 0.17/0.43 real_V55928688vector:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
% 0.17/0.43 execut510477386_space:
% 0.17/0.43 !>[A: $tType] : $o ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Affine_Oaff__dim,type,
% 0.17/0.43 aff_dim:
% 0.17/0.43 !>[A: $tType] : ( ( set @ A ) > int ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Fun_Obij__betw,type,
% 0.17/0.43 bij_betw:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Fun_Ocomp,type,
% 0.17/0.43 comp:
% 0.17/0.43 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Fun_Oid,type,
% 0.17/0.43 id:
% 0.17/0.43 !>[A: $tType] : ( A > A ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Fun_Oinj__on,type,
% 0.17/0.43 inj_on:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Fun_Oswap,type,
% 0.17/0.43 swap:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Fun_Othe__inv__into,type,
% 0.17/0.43 the_inv_into:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.17/0.43 zero_zero:
% 0.17/0.43 !>[A: $tType] : A ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 0.17/0.43 hilbert_inv_into:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner,type,
% 0.17/0.43 inner_780170721_inner:
% 0.17/0.43 !>[A: $tType] : ( A > A > real ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Linear__Algebra_Oadjoint,type,
% 0.17/0.43 linear_adjoint:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( A > B ) > B > A ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.17/0.43 top_top:
% 0.17/0.43 !>[A: $tType] : A ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Path__Connected_Oarc,type,
% 0.17/0.43 path_arc:
% 0.17/0.43 !>[A: $tType] : ( ( real > A ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Path__Connected_Opath,type,
% 0.17/0.43 path_path:
% 0.17/0.43 !>[A: $tType] : ( ( real > A ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Path__Connected_Opathfinish,type,
% 0.17/0.43 path_pathfinish:
% 0.17/0.43 !>[A: $tType] : ( ( real > A ) > A ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Path__Connected_Opathstart,type,
% 0.17/0.43 path_pathstart:
% 0.17/0.43 !>[A: $tType] : ( ( real > A ) > A ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Path__Connected_Orectpath,type,
% 0.17/0.43 path_rectpath: complex > complex > real > complex ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Path__Connected_Osimple__path,type,
% 0.17/0.43 path_simple_path:
% 0.17/0.43 !>[A: $tType] : ( ( real > A ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2,type,
% 0.17/0.43 poinca1076805489pen_R2:
% 0.17/0.43 !>[A: $tType] : ( ( A > A ) > ( A > ( bounde2145540817linfun @ A @ A ) ) > ( set @ A ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Ocomplex__of,type,
% 0.17/0.43 poinca1531302022lex_of:
% 0.17/0.43 !>[A: $tType] : ( A > complex ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Oreal__of,type,
% 0.17/0.43 poinca179318102eal_of:
% 0.17/0.43 !>[A: $tType] : ( complex > A ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
% 0.17/0.43 real_V1632203528linear:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
% 0.17/0.43 real_V1072672427axioms:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Real__Vector__Spaces_Olinear,type,
% 0.17/0.43 real_Vector_linear:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist,type,
% 0.17/0.43 iso_tu2011167877assist:
% 0.17/0.43 !>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > A > ( B > B ) > A > B > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Set_OCollect,type,
% 0.17/0.43 collect:
% 0.17/0.43 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Set_Oimage,type,
% 0.17/0.43 image:
% 0.17/0.43 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_Starlike_Ocoplanar,type,
% 0.17/0.43 coplanar:
% 0.17/0.43 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_c_member,type,
% 0.17/0.43 member:
% 0.17/0.43 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.17/0.43
% 0.17/0.43 thf(sy_v_c,type,
% 0.17/0.43 c: real > a ).
% 0.17/0.43
% 0.17/0.43 % Relevant facts (255)
% 0.17/0.43 thf(fact_0_assms_I2_J,axiom,
% 0.17/0.43 ( ( path_pathfinish @ a @ c )
% 0.17/0.43 = ( path_pathstart @ a @ c ) ) ).
% 0.17/0.43
% 0.17/0.43 % assms(2)
% 0.17/0.43 thf(fact_1_assms_I1_J,axiom,
% 0.17/0.43 path_simple_path @ a @ c ).
% 0.17/0.43
% 0.17/0.43 % assms(1)
% 0.17/0.43 thf(fact_2_a1,axiom,
% 0.17/0.43 path_simple_path @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) ).
% 0.17/0.43
% 0.17/0.43 % a1
% 0.17/0.43 thf(fact_3_comp__apply,axiom,
% 0.17/0.43 ! [C: $tType,A: $tType,B: $tType] :
% 0.17/0.43 ( ( comp @ B @ A @ C )
% 0.17/0.43 = ( ^ [F: B > A,G: C > B,X: C] : ( F @ ( G @ X ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_apply
% 0.17/0.43 thf(fact_4_pathfinish__compose,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType] :
% 0.17/0.43 ( ( ( topolo503727757_space @ B )
% 0.17/0.43 & ( topolo503727757_space @ A ) )
% 0.17/0.43 => ! [F2: B > A,P: real > B] :
% 0.17/0.43 ( ( path_pathfinish @ A @ ( comp @ B @ A @ real @ F2 @ P ) )
% 0.17/0.43 = ( F2 @ ( path_pathfinish @ B @ P ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pathfinish_compose
% 0.17/0.43 thf(fact_5_pathstart__compose,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType] :
% 0.17/0.43 ( ( ( topolo503727757_space @ B )
% 0.17/0.43 & ( topolo503727757_space @ A ) )
% 0.17/0.43 => ! [F2: B > A,P: real > B] :
% 0.17/0.43 ( ( path_pathstart @ A @ ( comp @ B @ A @ real @ F2 @ P ) )
% 0.17/0.43 = ( F2 @ ( path_pathstart @ B @ P ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pathstart_compose
% 0.17/0.43 thf(fact_6_complex__of__bounded__linear,axiom,
% 0.17/0.43 real_V1632203528linear @ a @ complex @ ( poinca1531302022lex_of @ a ) ).
% 0.17/0.43
% 0.17/0.43 % complex_of_bounded_linear
% 0.17/0.43 thf(fact_7_complex__of__linear,axiom,
% 0.17/0.43 real_Vector_linear @ a @ complex @ ( poinca1531302022lex_of @ a ) ).
% 0.17/0.43
% 0.17/0.43 % complex_of_linear
% 0.17/0.43 thf(fact_8_comp__def,axiom,
% 0.17/0.43 ! [A: $tType,C: $tType,B: $tType] :
% 0.17/0.43 ( ( comp @ B @ C @ A )
% 0.17/0.43 = ( ^ [F: B > C,G: A > B,X: A] : ( F @ ( G @ X ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_def
% 0.17/0.43 thf(fact_9_comp__assoc,axiom,
% 0.17/0.43 ! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H: A > C] :
% 0.17/0.43 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H )
% 0.17/0.43 = ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_assoc
% 0.17/0.43 thf(fact_10_comp__eq__dest,axiom,
% 0.17/0.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
% 0.17/0.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.17/0.43 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.17/0.43 => ( ( A2 @ ( B2 @ V ) )
% 0.17/0.43 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_eq_dest
% 0.17/0.43 thf(fact_11_comp__eq__elim,axiom,
% 0.17/0.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
% 0.17/0.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.17/0.43 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.17/0.43 => ! [V2: A] :
% 0.17/0.43 ( ( A2 @ ( B2 @ V2 ) )
% 0.17/0.43 = ( C2 @ ( D2 @ V2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_eq_elim
% 0.17/0.43 thf(fact_12_comp__cong,axiom,
% 0.17/0.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F2: B > A,G2: C > B,X2: C,F3: D > A,G3: E > D,X3: E] :
% 0.17/0.43 ( ( ( F2 @ ( G2 @ X2 ) )
% 0.17/0.43 = ( F3 @ ( G3 @ X3 ) ) )
% 0.17/0.43 => ( ( comp @ B @ A @ C @ F2 @ G2 @ X2 )
% 0.17/0.43 = ( comp @ D @ A @ E @ F3 @ G3 @ X3 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_cong
% 0.17/0.43 thf(fact_13_comp__eq__dest__lhs,axiom,
% 0.17/0.43 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 0.17/0.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.17/0.43 = C2 )
% 0.17/0.43 => ( ( A2 @ ( B2 @ V ) )
% 0.17/0.43 = ( C2 @ V ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_eq_dest_lhs
% 0.17/0.43 thf(fact_14_pathstart__linear__image__eq,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V1076094709vector @ A )
% 0.17/0.43 & ( topolo503727757_space @ A )
% 0.17/0.43 & ( real_V1076094709vector @ B )
% 0.17/0.43 & ( topolo503727757_space @ B ) )
% 0.17/0.43 => ! [F2: A > B,G2: real > A] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( path_pathstart @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 0.17/0.43 = ( F2 @ ( path_pathstart @ A @ G2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pathstart_linear_image_eq
% 0.17/0.43 thf(fact_15_pathfinish__linear__image,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V1076094709vector @ A )
% 0.17/0.43 & ( topolo503727757_space @ A )
% 0.17/0.43 & ( real_V1076094709vector @ B )
% 0.17/0.43 & ( topolo503727757_space @ B ) )
% 0.17/0.43 => ! [F2: A > B,G2: real > A] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( path_pathfinish @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 0.17/0.43 = ( F2 @ ( path_pathfinish @ A @ G2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pathfinish_linear_image
% 0.17/0.43 thf(fact_16_bounded__linear_Olinear,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V55928688vector @ A )
% 0.17/0.43 & ( real_V55928688vector @ B ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_V1632203528linear @ A @ B @ F2 )
% 0.17/0.43 => ( real_Vector_linear @ A @ B @ F2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bounded_linear.linear
% 0.17/0.43 thf(fact_17_linear__conv__bounded__linear,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( euclid925273238_space @ A )
% 0.17/0.43 & ( real_V55928688vector @ B ) )
% 0.17/0.43 => ( ( real_Vector_linear @ A @ B )
% 0.17/0.43 = ( real_V1632203528linear @ A @ B ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % linear_conv_bounded_linear
% 0.17/0.43 thf(fact_18_linear__compose,axiom,
% 0.17/0.43 ! [C: $tType,B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V1076094709vector @ A )
% 0.17/0.43 & ( real_V1076094709vector @ B )
% 0.17/0.43 & ( real_V1076094709vector @ C ) )
% 0.17/0.43 => ! [F2: A > B,G2: B > C] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( real_Vector_linear @ B @ C @ G2 )
% 0.17/0.43 => ( real_Vector_linear @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % linear_compose
% 0.17/0.43 thf(fact_19__092_060open_062_092_060And_062g_O_Ainj_Acomplex__of_A_092_060Longrightarrow_062_Asimple__path_A_Icomplex__of_A_092_060circ_062_Ag_J_A_061_Asimple__path_Ag_092_060close_062,axiom,
% 0.17/0.43 ! [G2: real > a] :
% 0.17/0.43 ( ( inj_on @ a @ complex @ ( poinca1531302022lex_of @ a ) @ ( top_top @ ( set @ a ) ) )
% 0.17/0.43 => ( ( path_simple_path @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ G2 ) )
% 0.17/0.43 = ( path_simple_path @ a @ G2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % \<open>\<And>g. inj complex_of \<Longrightarrow> simple_path (complex_of \<circ> g) = simple_path g\<close>
% 0.17/0.43 thf(fact_20_pathfinish__rectpath,axiom,
% 0.17/0.43 ! [A1: complex,A3: complex] :
% 0.17/0.43 ( ( path_pathfinish @ complex @ ( path_rectpath @ A1 @ A3 ) )
% 0.17/0.43 = A1 ) ).
% 0.17/0.43
% 0.17/0.43 % pathfinish_rectpath
% 0.17/0.43 thf(fact_21_pathstart__rectpath,axiom,
% 0.17/0.43 ! [A1: complex,A3: complex] :
% 0.17/0.43 ( ( path_pathstart @ complex @ ( path_rectpath @ A1 @ A3 ) )
% 0.17/0.43 = A1 ) ).
% 0.17/0.43
% 0.17/0.43 % pathstart_rectpath
% 0.17/0.43 thf(fact_22_bounded__linear_Obounded__linear,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V55928688vector @ A )
% 0.17/0.43 & ( real_V55928688vector @ B ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_V1632203528linear @ A @ B @ F2 )
% 0.17/0.43 => ( real_V1632203528linear @ A @ B @ F2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bounded_linear.bounded_linear
% 0.17/0.43 thf(fact_23_rewriteR__comp__comp2,axiom,
% 0.17/0.43 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R2: A > D,F2: B > E,L: D > E] :
% 0.17/0.43 ( ( ( comp @ C @ B @ A @ G2 @ H )
% 0.17/0.43 = ( comp @ D @ B @ A @ R1 @ R2 ) )
% 0.17/0.43 => ( ( ( comp @ B @ E @ D @ F2 @ R1 )
% 0.17/0.43 = L )
% 0.17/0.43 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F2 @ G2 ) @ H )
% 0.17/0.43 = ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % rewriteR_comp_comp2
% 0.17/0.43 thf(fact_24_rewriteL__comp__comp2,axiom,
% 0.17/0.43 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F2: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 0.17/0.43 ( ( ( comp @ C @ B @ A @ F2 @ G2 )
% 0.17/0.43 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.17/0.43 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 0.17/0.43 = R )
% 0.17/0.43 => ( ( comp @ C @ B @ E @ F2 @ ( comp @ A @ C @ E @ G2 @ H ) )
% 0.17/0.43 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % rewriteL_comp_comp2
% 0.17/0.43 thf(fact_25_rewriteR__comp__comp,axiom,
% 0.17/0.43 ! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R: A > B,F2: B > D] :
% 0.17/0.43 ( ( ( comp @ C @ B @ A @ G2 @ H )
% 0.17/0.43 = R )
% 0.17/0.43 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F2 @ G2 ) @ H )
% 0.17/0.43 = ( comp @ B @ D @ A @ F2 @ R ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % rewriteR_comp_comp
% 0.17/0.43 thf(fact_26_rewriteL__comp__comp,axiom,
% 0.17/0.43 ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: C > B,G2: A > C,L: A > B,H: D > A] :
% 0.17/0.43 ( ( ( comp @ C @ B @ A @ F2 @ G2 )
% 0.17/0.43 = L )
% 0.17/0.43 => ( ( comp @ C @ B @ D @ F2 @ ( comp @ A @ C @ D @ G2 @ H ) )
% 0.17/0.43 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % rewriteL_comp_comp
% 0.17/0.43 thf(fact_27_injD,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,X2: A,Y: A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( ( ( F2 @ X2 )
% 0.17/0.43 = ( F2 @ Y ) )
% 0.17/0.43 => ( X2 = Y ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % injD
% 0.17/0.43 thf(fact_28_injI,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B] :
% 0.17/0.43 ( ! [X4: A,Y2: A] :
% 0.17/0.43 ( ( ( F2 @ X4 )
% 0.17/0.43 = ( F2 @ Y2 ) )
% 0.17/0.43 => ( X4 = Y2 ) )
% 0.17/0.43 => ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % injI
% 0.17/0.43 thf(fact_29_inj__eq,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,X2: A,Y: A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( ( ( F2 @ X2 )
% 0.17/0.43 = ( F2 @ Y ) )
% 0.17/0.43 = ( X2 = Y ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_eq
% 0.17/0.43 thf(fact_30_inj__def,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 = ( ! [X: A,Y3: A] :
% 0.17/0.43 ( ( ( F2 @ X )
% 0.17/0.43 = ( F2 @ Y3 ) )
% 0.17/0.43 => ( X = Y3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_def
% 0.17/0.43 thf(fact_31_inj__onD,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.43 => ( ( ( F2 @ X2 )
% 0.17/0.43 = ( F2 @ Y ) )
% 0.17/0.43 => ( ( member @ A @ X2 @ A4 )
% 0.17/0.43 => ( ( member @ A @ Y @ A4 )
% 0.17/0.43 => ( X2 = Y ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_onD
% 0.17/0.43 thf(fact_32_inj__onI,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
% 0.17/0.43 ( ! [X4: A,Y2: A] :
% 0.17/0.43 ( ( member @ A @ X4 @ A4 )
% 0.17/0.43 => ( ( member @ A @ Y2 @ A4 )
% 0.17/0.43 => ( ( ( F2 @ X4 )
% 0.17/0.43 = ( F2 @ Y2 ) )
% 0.17/0.43 => ( X4 = Y2 ) ) ) )
% 0.17/0.43 => ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_onI
% 0.17/0.43 thf(fact_33_inj__on__def,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( inj_on @ A @ B )
% 0.17/0.43 = ( ^ [F: A > B,A5: set @ A] :
% 0.17/0.43 ! [X: A] :
% 0.17/0.43 ( ( member @ A @ X @ A5 )
% 0.17/0.43 => ! [Y3: A] :
% 0.17/0.43 ( ( member @ A @ Y3 @ A5 )
% 0.17/0.43 => ( ( ( F @ X )
% 0.17/0.43 = ( F @ Y3 ) )
% 0.17/0.43 => ( X = Y3 ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_def
% 0.17/0.43 thf(fact_34_inj__on__cong,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A4: set @ A,F2: A > B,G2: A > B] :
% 0.17/0.43 ( ! [A6: A] :
% 0.17/0.43 ( ( member @ A @ A6 @ A4 )
% 0.17/0.43 => ( ( F2 @ A6 )
% 0.17/0.43 = ( G2 @ A6 ) ) )
% 0.17/0.43 => ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.43 = ( inj_on @ A @ B @ G2 @ A4 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_cong
% 0.17/0.43 thf(fact_35_inj__on__eq__iff,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.43 => ( ( member @ A @ X2 @ A4 )
% 0.17/0.43 => ( ( member @ A @ Y @ A4 )
% 0.17/0.43 => ( ( ( F2 @ X2 )
% 0.17/0.43 = ( F2 @ Y ) )
% 0.17/0.43 = ( X2 = Y ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_eq_iff
% 0.17/0.43 thf(fact_36_inj__on__contraD,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.43 => ( ( X2 != Y )
% 0.17/0.43 => ( ( member @ A @ X2 @ A4 )
% 0.17/0.43 => ( ( member @ A @ Y @ A4 )
% 0.17/0.43 => ( ( F2 @ X2 )
% 0.17/0.43 != ( F2 @ Y ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_contraD
% 0.17/0.43 thf(fact_37_inj__on__inverseI,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A4: set @ A,G2: B > A,F2: A > B] :
% 0.17/0.43 ( ! [X4: A] :
% 0.17/0.43 ( ( member @ A @ X4 @ A4 )
% 0.17/0.43 => ( ( G2 @ ( F2 @ X4 ) )
% 0.17/0.43 = X4 ) )
% 0.17/0.43 => ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_inverseI
% 0.17/0.43 thf(fact_38_inj__compose,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: C > A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( ( inj_on @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) )
% 0.17/0.43 => ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F2 @ G2 ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_compose
% 0.17/0.43 thf(fact_39_inj__on__imageI2,axiom,
% 0.17/0.43 ! [B: $tType,C: $tType,A: $tType,F3: C > B,F2: A > C,A4: set @ A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F3 @ F2 ) @ A4 )
% 0.17/0.43 => ( inj_on @ A @ C @ F2 @ A4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_imageI2
% 0.17/0.43 thf(fact_40_simple__path__linear__image__eq,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( euclid925273238_space @ A )
% 0.17/0.43 & ( euclid925273238_space @ B ) )
% 0.17/0.43 => ! [F2: A > B,G2: real > A] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( ( path_simple_path @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 0.17/0.43 = ( path_simple_path @ A @ G2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % simple_path_linear_image_eq
% 0.17/0.43 thf(fact_41_fun_Oinj__map,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,D: $tType,F2: A > B] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F2 ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % fun.inj_map
% 0.17/0.43 thf(fact_42_real__of__bounded__linear,axiom,
% 0.17/0.43 real_V1632203528linear @ complex @ a @ ( poinca179318102eal_of @ a ) ).
% 0.17/0.43
% 0.17/0.43 % real_of_bounded_linear
% 0.17/0.43 thf(fact_43_real__of__linear,axiom,
% 0.17/0.43 real_Vector_linear @ complex @ a @ ( poinca179318102eal_of @ a ) ).
% 0.17/0.43
% 0.17/0.43 % real_of_linear
% 0.17/0.43 thf(fact_44_iso__tuple__UNIV__I,axiom,
% 0.17/0.43 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % iso_tuple_UNIV_I
% 0.17/0.43 thf(fact_45_mem__Collect__eq,axiom,
% 0.17/0.43 ! [A: $tType,A2: A,P2: A > $o] :
% 0.17/0.43 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.17/0.43 = ( P2 @ A2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % mem_Collect_eq
% 0.17/0.43 thf(fact_46_Collect__mem__eq,axiom,
% 0.17/0.43 ! [A: $tType,A4: set @ A] :
% 0.17/0.43 ( ( collect @ A
% 0.17/0.43 @ ^ [X: A] : ( member @ A @ X @ A4 ) )
% 0.17/0.43 = A4 ) ).
% 0.17/0.43
% 0.17/0.43 % Collect_mem_eq
% 0.17/0.43 thf(fact_47_Collect__cong,axiom,
% 0.17/0.43 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.17/0.43 ( ! [X4: A] :
% 0.17/0.43 ( ( P2 @ X4 )
% 0.17/0.43 = ( Q @ X4 ) )
% 0.17/0.43 => ( ( collect @ A @ P2 )
% 0.17/0.43 = ( collect @ A @ Q ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % Collect_cong
% 0.17/0.43 thf(fact_48_ext,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 0.17/0.43 ( ! [X4: A] :
% 0.17/0.43 ( ( F2 @ X4 )
% 0.17/0.43 = ( G2 @ X4 ) )
% 0.17/0.43 => ( F2 = G2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % ext
% 0.17/0.43 thf(fact_49_UNIV__I,axiom,
% 0.17/0.43 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % UNIV_I
% 0.17/0.43 thf(fact_50_top__apply,axiom,
% 0.17/0.43 ! [C: $tType,D: $tType] :
% 0.17/0.43 ( ( top @ C )
% 0.17/0.43 => ( ( top_top @ ( D > C ) )
% 0.17/0.43 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % top_apply
% 0.17/0.43 thf(fact_51_bounded__linear_Ointro,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V55928688vector @ A )
% 0.17/0.43 & ( real_V55928688vector @ B ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( real_V1072672427axioms @ A @ B @ F2 )
% 0.17/0.43 => ( real_V1632203528linear @ A @ B @ F2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bounded_linear.intro
% 0.17/0.43 thf(fact_52_bounded__linear__def,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V55928688vector @ A )
% 0.17/0.43 & ( real_V55928688vector @ B ) )
% 0.17/0.43 => ( ( real_V1632203528linear @ A @ B )
% 0.17/0.43 = ( ^ [F: A > B] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F )
% 0.17/0.43 & ( real_V1072672427axioms @ A @ B @ F ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bounded_linear_def
% 0.17/0.43 thf(fact_53_arc__linear__image__eq,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( euclid925273238_space @ A )
% 0.17/0.43 & ( euclid925273238_space @ B ) )
% 0.17/0.43 => ! [F2: A > B,G2: real > A] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( ( path_arc @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 0.17/0.43 = ( path_arc @ A @ G2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % arc_linear_image_eq
% 0.17/0.43 thf(fact_54_real__of__inj,axiom,
% 0.17/0.43 inj_on @ complex @ a @ ( poinca179318102eal_of @ a ) @ ( top_top @ ( set @ complex ) ) ).
% 0.17/0.43
% 0.17/0.43 % real_of_inj
% 0.17/0.43 thf(fact_55_top__set__def,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( top_top @ ( set @ A ) )
% 0.17/0.43 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % top_set_def
% 0.17/0.43 thf(fact_56_arc__imp__simple__path,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ! [G2: real > A] :
% 0.17/0.43 ( ( path_arc @ A @ G2 )
% 0.17/0.43 => ( path_simple_path @ A @ G2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % arc_imp_simple_path
% 0.17/0.43 thf(fact_57_arc__distinct__ends,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ! [G2: real > A] :
% 0.17/0.43 ( ( path_arc @ A @ G2 )
% 0.17/0.43 => ( ( path_pathfinish @ A @ G2 )
% 0.17/0.43 != ( path_pathstart @ A @ G2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % arc_distinct_ends
% 0.17/0.43 thf(fact_58_bounded__linear_Oaxioms_I2_J,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V55928688vector @ A )
% 0.17/0.43 & ( real_V55928688vector @ B ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_V1632203528linear @ A @ B @ F2 )
% 0.17/0.43 => ( real_V1072672427axioms @ A @ B @ F2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bounded_linear.axioms(2)
% 0.17/0.43 thf(fact_59_UNIV__eq__I,axiom,
% 0.17/0.43 ! [A: $tType,A4: set @ A] :
% 0.17/0.43 ( ! [X4: A] : ( member @ A @ X4 @ A4 )
% 0.17/0.43 => ( ( top_top @ ( set @ A ) )
% 0.17/0.43 = A4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % UNIV_eq_I
% 0.17/0.43 thf(fact_60_UNIV__witness,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % UNIV_witness
% 0.17/0.43 thf(fact_61_simple__path__imp__arc,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ! [G2: real > A] :
% 0.17/0.43 ( ( path_simple_path @ A @ G2 )
% 0.17/0.43 => ( ( ( path_pathfinish @ A @ G2 )
% 0.17/0.43 != ( path_pathstart @ A @ G2 ) )
% 0.17/0.43 => ( path_arc @ A @ G2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % simple_path_imp_arc
% 0.17/0.43 thf(fact_62_simple__path__eq__arc,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ! [G2: real > A] :
% 0.17/0.43 ( ( ( path_pathfinish @ A @ G2 )
% 0.17/0.43 != ( path_pathstart @ A @ G2 ) )
% 0.17/0.43 => ( ( path_simple_path @ A @ G2 )
% 0.17/0.43 = ( path_arc @ A @ G2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % simple_path_eq_arc
% 0.17/0.43 thf(fact_63_simple__path__cases,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ! [G2: real > A] :
% 0.17/0.43 ( ( path_simple_path @ A @ G2 )
% 0.17/0.43 => ( ( path_arc @ A @ G2 )
% 0.17/0.43 | ( ( path_pathfinish @ A @ G2 )
% 0.17/0.43 = ( path_pathstart @ A @ G2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % simple_path_cases
% 0.17/0.43 thf(fact_64_arc__simple__path,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ( ( path_arc @ A )
% 0.17/0.43 = ( ^ [G: real > A] :
% 0.17/0.43 ( ( path_simple_path @ A @ G )
% 0.17/0.43 & ( ( path_pathfinish @ A @ G )
% 0.17/0.43 != ( path_pathstart @ A @ G ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % arc_simple_path
% 0.17/0.43 thf(fact_65_fun_Omap__comp,axiom,
% 0.17/0.43 ! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
% 0.17/0.43 ( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
% 0.17/0.43 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% 0.17/0.43
% 0.17/0.43 % fun.map_comp
% 0.17/0.43 thf(fact_66_complex__of__real__of,axiom,
% 0.17/0.43 ( ( comp @ a @ complex @ complex @ ( poinca1531302022lex_of @ a ) @ ( poinca179318102eal_of @ a ) )
% 0.17/0.43 = ( id @ complex ) ) ).
% 0.17/0.43
% 0.17/0.43 % complex_of_real_of
% 0.17/0.43 thf(fact_67_real__of__complex__of,axiom,
% 0.17/0.43 ( ( comp @ complex @ a @ a @ ( poinca179318102eal_of @ a ) @ ( poinca1531302022lex_of @ a ) )
% 0.17/0.43 = ( id @ a ) ) ).
% 0.17/0.43
% 0.17/0.43 % real_of_complex_of
% 0.17/0.43 thf(fact_68_real__of__bij,axiom,
% 0.17/0.43 bij_betw @ complex @ a @ ( poinca179318102eal_of @ a ) @ ( top_top @ ( set @ complex ) ) @ ( top_top @ ( set @ a ) ) ).
% 0.17/0.43
% 0.17/0.43 % real_of_bij
% 0.17/0.43 thf(fact_69_complex__of__bij,axiom,
% 0.17/0.43 bij_betw @ a @ complex @ ( poinca1531302022lex_of @ a ) @ ( top_top @ ( set @ a ) ) @ ( top_top @ ( set @ complex ) ) ).
% 0.17/0.43
% 0.17/0.43 % complex_of_bij
% 0.17/0.43 thf(fact_70_linear__injective__left__inverse,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V1076094709vector @ A )
% 0.17/0.43 & ( real_V1076094709vector @ B ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ? [G4: B > A] :
% 0.17/0.43 ( ( real_Vector_linear @ B @ A @ G4 )
% 0.17/0.43 & ( ( comp @ B @ A @ A @ G4 @ F2 )
% 0.17/0.43 = ( id @ A ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % linear_injective_left_inverse
% 0.17/0.43 thf(fact_71_path__linear__image__eq,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( euclid925273238_space @ A )
% 0.17/0.43 & ( euclid925273238_space @ B ) )
% 0.17/0.43 => ! [F2: A > B,G2: real > A] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( ( path_path @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 0.17/0.43 = ( path_path @ A @ G2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % path_linear_image_eq
% 0.17/0.43 thf(fact_72_linear__inj__iff__eq__0,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( ( real_V1076094709vector @ A )
% 0.17/0.43 & ( real_V1076094709vector @ B ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 = ( ! [X: A] :
% 0.17/0.43 ( ( ( F2 @ X )
% 0.17/0.43 = ( zero_zero @ B ) )
% 0.17/0.43 => ( X
% 0.17/0.43 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % linear_inj_iff_eq_0
% 0.17/0.43 thf(fact_73_type__copy__map__cong0,axiom,
% 0.17/0.43 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G2: C > B,X2: C,N: D > A,H: C > D,F2: A > E] :
% 0.17/0.43 ( ( ( M @ ( G2 @ X2 ) )
% 0.17/0.43 = ( N @ ( H @ X2 ) ) )
% 0.17/0.43 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F2 @ M ) @ G2 @ X2 )
% 0.17/0.43 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F2 @ N ) @ H @ X2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % type_copy_map_cong0
% 0.17/0.43 thf(fact_74_id__apply,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( id @ A )
% 0.17/0.43 = ( ^ [X: A] : X ) ) ).
% 0.17/0.43
% 0.17/0.43 % id_apply
% 0.17/0.43 thf(fact_75_top1I,axiom,
% 0.17/0.43 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 0.17/0.43
% 0.17/0.43 % top1I
% 0.17/0.43 thf(fact_76_path__rectpath,axiom,
% 0.17/0.43 ! [A2: complex,B2: complex] : ( path_path @ complex @ ( path_rectpath @ A2 @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % path_rectpath
% 0.17/0.43 thf(fact_77_comp__id,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B] :
% 0.17/0.43 ( ( comp @ A @ B @ A @ F2 @ ( id @ A ) )
% 0.17/0.43 = F2 ) ).
% 0.17/0.43
% 0.17/0.43 % comp_id
% 0.17/0.43 thf(fact_78_id__comp,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,G2: A > B] :
% 0.17/0.43 ( ( comp @ B @ B @ A @ ( id @ B ) @ G2 )
% 0.17/0.43 = G2 ) ).
% 0.17/0.43
% 0.17/0.43 % id_comp
% 0.17/0.43 thf(fact_79_fun_Omap__id,axiom,
% 0.17/0.43 ! [A: $tType,D: $tType,T: D > A] :
% 0.17/0.43 ( ( comp @ A @ A @ D @ ( id @ A ) @ T )
% 0.17/0.43 = T ) ).
% 0.17/0.43
% 0.17/0.43 % fun.map_id
% 0.17/0.43 thf(fact_80_bij__betw__id,axiom,
% 0.17/0.43 ! [A: $tType,A4: set @ A] : ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ A4 ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_id
% 0.17/0.43 thf(fact_81_bij__id,axiom,
% 0.17/0.43 ! [A: $tType] : ( bij_betw @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_id
% 0.17/0.43 thf(fact_82_DEADID_Oin__rel,axiom,
% 0.17/0.43 ! [B: $tType] :
% 0.17/0.43 ( ( ^ [Y4: B,Z: B] : ( Y4 = Z ) )
% 0.17/0.43 = ( ^ [A7: B,B3: B] :
% 0.17/0.43 ? [Z2: B] :
% 0.17/0.43 ( ( member @ B @ Z2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 & ( ( id @ B @ Z2 )
% 0.17/0.43 = A7 )
% 0.17/0.43 & ( ( id @ B @ Z2 )
% 0.17/0.43 = B3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % DEADID.in_rel
% 0.17/0.43 thf(fact_83_top__conj_I1_J,axiom,
% 0.17/0.43 ! [A: $tType,X2: A,P2: $o] :
% 0.17/0.43 ( ( ( top_top @ ( A > $o ) @ X2 )
% 0.17/0.43 & P2 )
% 0.17/0.43 = P2 ) ).
% 0.17/0.43
% 0.17/0.43 % top_conj(1)
% 0.17/0.43 thf(fact_84_top__conj_I2_J,axiom,
% 0.17/0.43 ! [A: $tType,P2: $o,X2: A] :
% 0.17/0.43 ( ( P2
% 0.17/0.43 & ( top_top @ ( A > $o ) @ X2 ) )
% 0.17/0.43 = P2 ) ).
% 0.17/0.43
% 0.17/0.43 % top_conj(2)
% 0.17/0.43 thf(fact_85_fun_Omap__id0,axiom,
% 0.17/0.43 ! [A: $tType,D: $tType] :
% 0.17/0.43 ( ( comp @ A @ A @ D @ ( id @ A ) )
% 0.17/0.43 = ( id @ ( D > A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % fun.map_id0
% 0.17/0.43 thf(fact_86_id__def,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( id @ A )
% 0.17/0.43 = ( ^ [X: A] : X ) ) ).
% 0.17/0.43
% 0.17/0.43 % id_def
% 0.17/0.43 thf(fact_87_bij__betwE,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.43 => ! [X5: A] :
% 0.17/0.43 ( ( member @ A @ X5 @ A4 )
% 0.17/0.43 => ( member @ B @ ( F2 @ X5 ) @ B4 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betwE
% 0.17/0.43 thf(fact_88_eq__id__iff,axiom,
% 0.17/0.43 ! [A: $tType,F2: A > A] :
% 0.17/0.43 ( ( ! [X: A] :
% 0.17/0.43 ( ( F2 @ X )
% 0.17/0.43 = X ) )
% 0.17/0.43 = ( F2
% 0.17/0.43 = ( id @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % eq_id_iff
% 0.17/0.43 thf(fact_89_bij__betw__inv,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.43 => ? [G4: B > A] : ( bij_betw @ B @ A @ G4 @ B4 @ A4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_inv
% 0.17/0.43 thf(fact_90_bij__betw__cong,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,A4: set @ A,F2: A > B,G2: A > B,A8: set @ B] :
% 0.17/0.43 ( ! [A6: A] :
% 0.17/0.43 ( ( member @ A @ A6 @ A4 )
% 0.17/0.43 => ( ( F2 @ A6 )
% 0.17/0.43 = ( G2 @ A6 ) ) )
% 0.17/0.43 => ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 0.17/0.43 = ( bij_betw @ A @ B @ G2 @ A4 @ A8 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_cong
% 0.17/0.43 thf(fact_91_bij__betw__apply,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B,A2: A] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.43 => ( ( member @ A @ A2 @ A4 )
% 0.17/0.43 => ( member @ B @ ( F2 @ A2 ) @ B4 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_apply
% 0.17/0.43 thf(fact_92_bij__betw__id__iff,axiom,
% 0.17/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.17/0.43 ( ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ B4 )
% 0.17/0.43 = ( A4 = B4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_id_iff
% 0.17/0.43 thf(fact_93_bij__betw__iff__bijections,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( bij_betw @ A @ B )
% 0.17/0.43 = ( ^ [F: A > B,A5: set @ A,B5: set @ B] :
% 0.17/0.43 ? [G: B > A] :
% 0.17/0.43 ( ! [X: A] :
% 0.17/0.43 ( ( member @ A @ X @ A5 )
% 0.17/0.43 => ( ( member @ B @ ( F @ X ) @ B5 )
% 0.17/0.43 & ( ( G @ ( F @ X ) )
% 0.17/0.43 = X ) ) )
% 0.17/0.43 & ! [X: B] :
% 0.17/0.43 ( ( member @ B @ X @ B5 )
% 0.17/0.43 => ( ( member @ A @ ( G @ X ) @ A5 )
% 0.17/0.43 & ( ( F @ ( G @ X ) )
% 0.17/0.43 = X ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_iff_bijections
% 0.17/0.43 thf(fact_94_o__bij,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,G2: B > A,F2: A > B] :
% 0.17/0.43 ( ( ( comp @ B @ A @ A @ G2 @ F2 )
% 0.17/0.43 = ( id @ A ) )
% 0.17/0.43 => ( ( ( comp @ A @ B @ B @ F2 @ G2 )
% 0.17/0.43 = ( id @ B ) )
% 0.17/0.43 => ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % o_bij
% 0.17/0.43 thf(fact_95_bij__pointE,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,Y: B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 => ~ ! [X4: A] :
% 0.17/0.43 ( ( Y
% 0.17/0.43 = ( F2 @ X4 ) )
% 0.17/0.43 => ~ ! [X6: A] :
% 0.17/0.43 ( ( Y
% 0.17/0.43 = ( F2 @ X6 ) )
% 0.17/0.43 => ( X6 = X4 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_pointE
% 0.17/0.43 thf(fact_96_bij__betw__trans,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,C: $tType,F2: A > B,A4: set @ A,B4: set @ B,G2: B > C,C3: set @ C] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.43 => ( ( bij_betw @ B @ C @ G2 @ B4 @ C3 )
% 0.17/0.43 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ A4 @ C3 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_trans
% 0.17/0.43 thf(fact_97_bij__betw__comp__iff,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,C: $tType,F2: A > B,A4: set @ A,A8: set @ B,F3: B > C,A9: set @ C] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 0.17/0.43 => ( ( bij_betw @ B @ C @ F3 @ A8 @ A9 )
% 0.17/0.43 = ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F3 @ F2 ) @ A4 @ A9 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_comp_iff
% 0.17/0.43 thf(fact_98_bij__betw__imp__inj__on,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.43 => ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_imp_inj_on
% 0.17/0.43 thf(fact_99_pointfree__idE,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: B > A,G2: A > B,X2: A] :
% 0.17/0.43 ( ( ( comp @ B @ A @ A @ F2 @ G2 )
% 0.17/0.43 = ( id @ A ) )
% 0.17/0.43 => ( ( F2 @ ( G2 @ X2 ) )
% 0.17/0.43 = X2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % pointfree_idE
% 0.17/0.43 thf(fact_100_comp__eq__id__dest,axiom,
% 0.17/0.43 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 0.17/0.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.17/0.43 = ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
% 0.17/0.43 => ( ( A2 @ ( B2 @ V ) )
% 0.17/0.43 = ( C2 @ V ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_eq_id_dest
% 0.17/0.43 thf(fact_101_arc__imp__path,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ! [G2: real > A] :
% 0.17/0.43 ( ( path_arc @ A @ G2 )
% 0.17/0.43 => ( path_path @ A @ G2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % arc_imp_path
% 0.17/0.43 thf(fact_102_inj__on__id,axiom,
% 0.17/0.43 ! [A: $tType,A4: set @ A] : ( inj_on @ A @ A @ ( id @ A ) @ A4 ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_id
% 0.17/0.43 thf(fact_103_simple__path__imp__path,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ! [G2: real > A] :
% 0.17/0.43 ( ( path_simple_path @ A @ G2 )
% 0.17/0.43 => ( path_path @ A @ G2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % simple_path_imp_path
% 0.17/0.43 thf(fact_104_module__hom__id,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( real_V1076094709vector @ A )
% 0.17/0.43 => ( real_Vector_linear @ A @ A @ ( id @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % module_hom_id
% 0.17/0.43 thf(fact_105_linear__simps_I3_J,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType] :
% 0.17/0.43 ( ( ( real_V55928688vector @ B )
% 0.17/0.43 & ( real_V55928688vector @ A ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_V1632203528linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( F2 @ ( zero_zero @ A ) )
% 0.17/0.43 = ( zero_zero @ B ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % linear_simps(3)
% 0.17/0.43 thf(fact_106_linear__0,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType] :
% 0.17/0.43 ( ( ( real_V1076094709vector @ B )
% 0.17/0.43 & ( real_V1076094709vector @ A ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( F2 @ ( zero_zero @ A ) )
% 0.17/0.43 = ( zero_zero @ B ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % linear_0
% 0.17/0.43 thf(fact_107_pathstart__def,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( topolo503727757_space @ A )
% 0.17/0.43 => ( ( path_pathstart @ A )
% 0.17/0.43 = ( ^ [G: real > A] : ( G @ ( zero_zero @ real ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pathstart_def
% 0.17/0.43 thf(fact_108_bij__comp,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: B > C] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 => ( ( bij_betw @ B @ C @ G2 @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ C ) ) )
% 0.17/0.43 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_comp
% 0.17/0.43 thf(fact_109_bij__is__inj,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 => ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_is_inj
% 0.17/0.43 thf(fact_110_right__inverse__linear,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( euclid925273238_space @ A )
% 0.17/0.43 => ! [F2: A > A,G2: A > A] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ A @ F2 )
% 0.17/0.43 => ( ( ( comp @ A @ A @ A @ F2 @ G2 )
% 0.17/0.43 = ( id @ A ) )
% 0.17/0.43 => ( real_Vector_linear @ A @ A @ G2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % right_inverse_linear
% 0.17/0.43 thf(fact_111_linear__inverse__left,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( euclid925273238_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > A] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ A @ F2 )
% 0.17/0.43 => ( ( real_Vector_linear @ A @ A @ F3 )
% 0.17/0.43 => ( ( ( comp @ A @ A @ A @ F2 @ F3 )
% 0.17/0.43 = ( id @ A ) )
% 0.17/0.43 = ( ( comp @ A @ A @ A @ F3 @ F2 )
% 0.17/0.43 = ( id @ A ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % linear_inverse_left
% 0.17/0.43 thf(fact_112_left__inverse__linear,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( euclid925273238_space @ A )
% 0.17/0.43 => ! [F2: A > A,G2: A > A] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ A @ F2 )
% 0.17/0.43 => ( ( ( comp @ A @ A @ A @ G2 @ F2 )
% 0.17/0.43 = ( id @ A ) )
% 0.17/0.43 => ( real_Vector_linear @ A @ A @ G2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % left_inverse_linear
% 0.17/0.43 thf(fact_113_top__empty__eq,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( top_top @ ( A > $o ) )
% 0.17/0.43 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % top_empty_eq
% 0.17/0.43 thf(fact_114_left__right__inverse__eq,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,F2: B > A,G2: A > B,H: B > A] :
% 0.17/0.43 ( ( ( comp @ B @ A @ A @ F2 @ G2 )
% 0.17/0.43 = ( id @ A ) )
% 0.17/0.43 => ( ( ( comp @ A @ B @ B @ G2 @ H )
% 0.17/0.43 = ( id @ B ) )
% 0.17/0.43 => ( F2 = H ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % left_right_inverse_eq
% 0.17/0.43 thf(fact_115_zero__reorient,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( zero @ A )
% 0.17/0.43 => ! [X2: A] :
% 0.17/0.43 ( ( ( zero_zero @ A )
% 0.17/0.43 = X2 )
% 0.17/0.43 = ( X2
% 0.17/0.43 = ( zero_zero @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_reorient
% 0.17/0.43 thf(fact_116_isomorphism__expand,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,F2: B > A,G2: A > B] :
% 0.17/0.43 ( ( ( ( comp @ B @ A @ A @ F2 @ G2 )
% 0.17/0.43 = ( id @ A ) )
% 0.17/0.43 & ( ( comp @ A @ B @ B @ G2 @ F2 )
% 0.17/0.43 = ( id @ B ) ) )
% 0.17/0.43 = ( ! [X: A] :
% 0.17/0.43 ( ( F2 @ ( G2 @ X ) )
% 0.17/0.43 = X )
% 0.17/0.43 & ! [X: B] :
% 0.17/0.43 ( ( G2 @ ( F2 @ X ) )
% 0.17/0.43 = X ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % isomorphism_expand
% 0.17/0.43 thf(fact_117_bijI_H,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,F2: A > B] :
% 0.17/0.43 ( ! [X4: A,Y2: A] :
% 0.17/0.43 ( ( ( F2 @ X4 )
% 0.17/0.43 = ( F2 @ Y2 ) )
% 0.17/0.43 = ( X4 = Y2 ) )
% 0.17/0.43 => ( ! [Y2: B] :
% 0.17/0.43 ? [X5: A] :
% 0.17/0.43 ( Y2
% 0.17/0.43 = ( F2 @ X5 ) )
% 0.17/0.43 => ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bijI'
% 0.17/0.43 thf(fact_118_bij__iff,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,F2: A > B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 = ( ! [X: B] :
% 0.17/0.43 ? [Y3: A] :
% 0.17/0.43 ( ( ( F2 @ Y3 )
% 0.17/0.43 = X )
% 0.17/0.43 & ! [Z2: A] :
% 0.17/0.43 ( ( ( F2 @ Z2 )
% 0.17/0.43 = X )
% 0.17/0.43 => ( Z2 = Y3 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_iff
% 0.17/0.43 thf(fact_119_c1__on__open__R2_Ocomplex__of__real__of,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( ( comp @ A @ complex @ complex @ ( poinca1531302022lex_of @ A ) @ ( poinca179318102eal_of @ A ) )
% 0.17/0.43 = ( id @ complex ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.complex_of_real_of
% 0.17/0.43 thf(fact_120_c1__on__open__R2_Oreal__of__bij,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( bij_betw @ complex @ A @ ( poinca179318102eal_of @ A ) @ ( top_top @ ( set @ complex ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.real_of_bij
% 0.17/0.43 thf(fact_121_c1__on__open__R2_Oreal__of__bounded__linear,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( real_V1632203528linear @ complex @ A @ ( poinca179318102eal_of @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.real_of_bounded_linear
% 0.17/0.43 thf(fact_122_c1__on__open__R2_Ocomplex__of__bounded__linear,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( real_V1632203528linear @ A @ complex @ ( poinca1531302022lex_of @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.complex_of_bounded_linear
% 0.17/0.43 thf(fact_123_c1__on__open__R2_Ocomplex__of__linear,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( real_Vector_linear @ A @ complex @ ( poinca1531302022lex_of @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.complex_of_linear
% 0.17/0.43 thf(fact_124_c1__on__open__R2_Oreal__of__linear,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( real_Vector_linear @ complex @ A @ ( poinca179318102eal_of @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.real_of_linear
% 0.17/0.43 thf(fact_125_comp__apply__eq,axiom,
% 0.17/0.43 ! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,X2: C,H: D > A,K: C > D] :
% 0.17/0.43 ( ( ( F2 @ ( G2 @ X2 ) )
% 0.17/0.43 = ( H @ ( K @ X2 ) ) )
% 0.17/0.43 => ( ( comp @ B @ A @ C @ F2 @ G2 @ X2 )
% 0.17/0.43 = ( comp @ D @ A @ C @ H @ K @ X2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_apply_eq
% 0.17/0.43 thf(fact_126_c1__on__open__R2_Oreal__of__inj,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( inj_on @ complex @ A @ ( poinca179318102eal_of @ A ) @ ( top_top @ ( set @ complex ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.real_of_inj
% 0.17/0.43 thf(fact_127_c1__on__open__R2_Oreal__of__complex__of,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( ( comp @ complex @ A @ A @ ( poinca179318102eal_of @ A ) @ ( poinca1531302022lex_of @ A ) )
% 0.17/0.43 = ( id @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.real_of_complex_of
% 0.17/0.43 thf(fact_128_c1__on__open__R2_Ocomplex__of__bij,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( execut510477386_space @ A )
% 0.17/0.43 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 0.17/0.43 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 0.17/0.43 => ( bij_betw @ A @ complex @ ( poinca1531302022lex_of @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ complex ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % c1_on_open_R2.complex_of_bij
% 0.17/0.43 thf(fact_129_the__inv__f__o__f__id,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,Z3: A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 @ Z3 )
% 0.17/0.43 = ( id @ A @ Z3 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % the_inv_f_o_f_id
% 0.17/0.43 thf(fact_130_linear__surjective__right__inverse,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType] :
% 0.17/0.43 ( ( ( real_V1076094709vector @ B )
% 0.17/0.43 & ( real_V1076094709vector @ A ) )
% 0.17/0.43 => ! [F2: A > B] :
% 0.17/0.43 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.43 => ( ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.43 = ( top_top @ ( set @ B ) ) )
% 0.17/0.43 => ? [G4: B > A] :
% 0.17/0.43 ( ( real_Vector_linear @ B @ A @ G4 )
% 0.17/0.43 & ( ( comp @ A @ B @ B @ F2 @ G4 )
% 0.17/0.43 = ( id @ B ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % linear_surjective_right_inverse
% 0.17/0.43 thf(fact_131_iso__tuple__update__accessor__eq__assist__idI,axiom,
% 0.17/0.43 ! [A: $tType,V3: A,F2: A > A,V: A] :
% 0.17/0.43 ( ( V3
% 0.17/0.43 = ( F2 @ V ) )
% 0.17/0.43 => ( iso_tu2011167877assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) @ V @ F2 @ V3 @ V ) ) ).
% 0.17/0.43
% 0.17/0.43 % iso_tuple_update_accessor_eq_assist_idI
% 0.17/0.43 thf(fact_132_bij__swap__compose__bij,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,P: A > B,A2: B,B2: B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 => ( bij_betw @ A @ B @ ( comp @ B @ B @ A @ ( swap @ B @ B @ A2 @ B2 @ ( id @ B ) ) @ P ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_swap_compose_bij
% 0.17/0.43 thf(fact_133_image__eqI,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,B2: A,F2: B > A,X2: B,A4: set @ B] :
% 0.17/0.43 ( ( B2
% 0.17/0.43 = ( F2 @ X2 ) )
% 0.17/0.43 => ( ( member @ B @ X2 @ A4 )
% 0.17/0.43 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % image_eqI
% 0.17/0.43 thf(fact_134_swap__nilpotent,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A2: A,B2: A,F2: A > B] :
% 0.17/0.43 ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ A2 @ B2 @ F2 ) )
% 0.17/0.43 = F2 ) ).
% 0.17/0.43
% 0.17/0.43 % swap_nilpotent
% 0.17/0.43 thf(fact_135_swap__self,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A2: A,F2: A > B] :
% 0.17/0.43 ( ( swap @ A @ B @ A2 @ A2 @ F2 )
% 0.17/0.43 = F2 ) ).
% 0.17/0.43
% 0.17/0.43 % swap_self
% 0.17/0.43 thf(fact_136_swap__apply_I3_J,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,C2: B,A2: B,B2: B,F2: B > A] :
% 0.17/0.43 ( ( C2 != A2 )
% 0.17/0.43 => ( ( C2 != B2 )
% 0.17/0.43 => ( ( swap @ B @ A @ A2 @ B2 @ F2 @ C2 )
% 0.17/0.43 = ( F2 @ C2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % swap_apply(3)
% 0.17/0.43 thf(fact_137_image__id,axiom,
% 0.17/0.43 ! [A: $tType] :
% 0.17/0.43 ( ( image @ A @ A @ ( id @ A ) )
% 0.17/0.43 = ( id @ ( set @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % image_id
% 0.17/0.43 thf(fact_138_swap__image__eq,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: A,F2: A > B] :
% 0.17/0.43 ( ( member @ A @ A2 @ A4 )
% 0.17/0.43 => ( ( member @ A @ B2 @ A4 )
% 0.17/0.43 => ( ( image @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 )
% 0.17/0.43 = ( image @ A @ B @ F2 @ A4 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % swap_image_eq
% 0.17/0.43 thf(fact_139_inj__on__swap__iff,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: A,F2: A > B] :
% 0.17/0.43 ( ( member @ A @ A2 @ A4 )
% 0.17/0.43 => ( ( member @ A @ B2 @ A4 )
% 0.17/0.43 => ( ( inj_on @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 )
% 0.17/0.43 = ( inj_on @ A @ B @ F2 @ A4 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_swap_iff
% 0.17/0.43 thf(fact_140_bij__betw__swap__iff,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,X2: A,A4: set @ A,Y: A,F2: A > B,B4: set @ B] :
% 0.17/0.43 ( ( member @ A @ X2 @ A4 )
% 0.17/0.43 => ( ( member @ A @ Y @ A4 )
% 0.17/0.43 => ( ( bij_betw @ A @ B @ ( swap @ A @ B @ X2 @ Y @ F2 ) @ A4 @ B4 )
% 0.17/0.43 = ( bij_betw @ A @ B @ F2 @ A4 @ B4 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_swap_iff
% 0.17/0.43 thf(fact_141_swap__comp__involutory,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A2: A,B2: A] :
% 0.17/0.43 ( ( comp @ ( A > B ) @ ( A > B ) @ ( A > B ) @ ( swap @ A @ B @ A2 @ B2 ) @ ( swap @ A @ B @ A2 @ B2 ) )
% 0.17/0.43 = ( id @ ( A > B ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % swap_comp_involutory
% 0.17/0.43 thf(fact_142_surj__swap__iff,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A2: B,B2: B,F2: B > A] :
% 0.17/0.43 ( ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 = ( top_top @ ( set @ A ) ) )
% 0.17/0.43 = ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % surj_swap_iff
% 0.17/0.43 thf(fact_143_bij__swap__iff,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,A2: A,B2: A,F2: A > B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 = ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_swap_iff
% 0.17/0.43 thf(fact_144_swap__id__idempotent,axiom,
% 0.17/0.43 ! [A: $tType,A2: A,B2: A] :
% 0.17/0.43 ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) )
% 0.17/0.43 = ( id @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % swap_id_idempotent
% 0.17/0.43 thf(fact_145_the__inv__into__onto,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.43 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ ( image @ A @ B @ F2 @ A4 ) )
% 0.17/0.43 = A4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % the_inv_into_onto
% 0.17/0.43 thf(fact_146_swap__id__eq,axiom,
% 0.17/0.43 ! [A: $tType,X2: A,A2: A,B2: A] :
% 0.17/0.43 ( ( ( X2 = A2 )
% 0.17/0.43 => ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
% 0.17/0.43 = B2 ) )
% 0.17/0.43 & ( ( X2 != A2 )
% 0.17/0.43 => ( ( ( X2 = B2 )
% 0.17/0.43 => ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
% 0.17/0.43 = A2 ) )
% 0.17/0.43 & ( ( X2 != B2 )
% 0.17/0.43 => ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
% 0.17/0.43 = X2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % swap_id_eq
% 0.17/0.43 thf(fact_147_comp__swap,axiom,
% 0.17/0.43 ! [B: $tType,C: $tType,A: $tType,F2: C > B,A2: A,B2: A,G2: A > C] :
% 0.17/0.43 ( ( comp @ C @ B @ A @ F2 @ ( swap @ A @ C @ A2 @ B2 @ G2 ) )
% 0.17/0.43 = ( swap @ A @ B @ A2 @ B2 @ ( comp @ C @ B @ A @ F2 @ G2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % comp_swap
% 0.17/0.43 thf(fact_148_the__inv__into__comp,axiom,
% 0.17/0.43 ! [A: $tType,C: $tType,B: $tType,F2: A > B,G2: C > A,A4: set @ C,X2: B] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ ( image @ C @ A @ G2 @ A4 ) )
% 0.17/0.43 => ( ( inj_on @ C @ A @ G2 @ A4 )
% 0.17/0.43 => ( ( member @ B @ X2 @ ( image @ A @ B @ F2 @ ( image @ C @ A @ G2 @ A4 ) ) )
% 0.17/0.43 => ( ( the_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F2 @ G2 ) @ X2 )
% 0.17/0.43 = ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A4 @ G2 ) @ ( the_inv_into @ A @ B @ ( image @ C @ A @ G2 @ A4 ) @ F2 ) @ X2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % the_inv_into_comp
% 0.17/0.43 thf(fact_149_image__eq__imp__comp,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,D: $tType,C: $tType,F2: B > A,A4: set @ B,G2: C > A,B4: set @ C,H: A > D] :
% 0.17/0.43 ( ( ( image @ B @ A @ F2 @ A4 )
% 0.17/0.43 = ( image @ C @ A @ G2 @ B4 ) )
% 0.17/0.43 => ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F2 ) @ A4 )
% 0.17/0.43 = ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G2 ) @ B4 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % image_eq_imp_comp
% 0.17/0.43 thf(fact_150_image__comp,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,R: set @ C] :
% 0.17/0.43 ( ( image @ B @ A @ F2 @ ( image @ C @ B @ G2 @ R ) )
% 0.17/0.43 = ( image @ C @ A @ ( comp @ B @ A @ C @ F2 @ G2 ) @ R ) ) ).
% 0.17/0.43
% 0.17/0.43 % image_comp
% 0.17/0.43 thf(fact_151_bij__betw__imp__surj__on,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 0.17/0.43 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.43 => ( ( image @ A @ B @ F2 @ A4 )
% 0.17/0.43 = B4 ) ) ).
% 0.17/0.43
% 0.17/0.43 % bij_betw_imp_surj_on
% 0.17/0.43 thf(fact_152_surj__imp__surj__swap,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: B > A,A2: B,B2: B] :
% 0.17/0.43 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 = ( top_top @ ( set @ A ) ) )
% 0.17/0.43 => ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.43 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % surj_imp_surj_swap
% 0.17/0.43 thf(fact_153_inj__on__the__inv__into,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.43 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % inj_on_the_inv_into
% 0.17/0.43 thf(fact_154_f__the__inv__into__f,axiom,
% 0.17/0.43 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,Y: B] :
% 0.17/0.43 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.43 => ( ( member @ B @ Y @ ( image @ A @ B @ F2 @ A4 ) )
% 0.17/0.43 => ( ( F2 @ ( the_inv_into @ A @ B @ A4 @ F2 @ Y ) )
% 0.17/0.43 = Y ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % f_the_inv_into_f
% 0.17/0.43 thf(fact_155_swap__commute,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType] :
% 0.17/0.43 ( ( swap @ A @ B )
% 0.17/0.43 = ( ^ [A7: A,B3: A] : ( swap @ A @ B @ B3 @ A7 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % swap_commute
% 0.17/0.43 thf(fact_156_swap__triple,axiom,
% 0.17/0.43 ! [B: $tType,A: $tType,A2: A,C2: A,B2: A,F2: A > B] :
% 0.17/0.43 ( ( A2 != C2 )
% 0.17/0.43 => ( ( B2 != C2 )
% 0.17/0.44 => ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ B2 @ C2 @ ( swap @ A @ B @ A2 @ B2 @ F2 ) ) )
% 0.17/0.44 = ( swap @ A @ B @ A2 @ C2 @ F2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % swap_triple
% 0.17/0.44 thf(fact_157_iso__tuple__update__accessor__eq__assist__triv,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
% 0.17/0.44 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
% 0.17/0.44 => ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % iso_tuple_update_accessor_eq_assist_triv
% 0.17/0.44 thf(fact_158_update__accessor__accessor__eqE,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
% 0.17/0.44 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
% 0.17/0.44 => ( ( Ac @ V )
% 0.17/0.44 = X2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % update_accessor_accessor_eqE
% 0.17/0.44 thf(fact_159_update__accessor__updator__eqE,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
% 0.17/0.44 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
% 0.17/0.44 => ( ( Upd @ F2 @ V )
% 0.17/0.44 = V3 ) ) ).
% 0.17/0.44
% 0.17/0.44 % update_accessor_updator_eqE
% 0.17/0.44 thf(fact_160_rev__image__eqI,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,X2: A,A4: set @ A,B2: B,F2: A > B] :
% 0.17/0.44 ( ( member @ A @ X2 @ A4 )
% 0.17/0.44 => ( ( B2
% 0.17/0.44 = ( F2 @ X2 ) )
% 0.17/0.44 => ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % rev_image_eqI
% 0.17/0.44 thf(fact_161_ball__imageD,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P2: A > $o] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( member @ A @ X4 @ ( image @ B @ A @ F2 @ A4 ) )
% 0.17/0.44 => ( P2 @ X4 ) )
% 0.17/0.44 => ! [X5: B] :
% 0.17/0.44 ( ( member @ B @ X5 @ A4 )
% 0.17/0.44 => ( P2 @ ( F2 @ X5 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ball_imageD
% 0.17/0.44 thf(fact_162_image__cong,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F2: A > B,G2: A > B] :
% 0.17/0.44 ( ( M = N )
% 0.17/0.44 => ( ! [X4: A] :
% 0.17/0.44 ( ( member @ A @ X4 @ N )
% 0.17/0.44 => ( ( F2 @ X4 )
% 0.17/0.44 = ( G2 @ X4 ) ) )
% 0.17/0.44 => ( ( image @ A @ B @ F2 @ M )
% 0.17/0.44 = ( image @ A @ B @ G2 @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % image_cong
% 0.17/0.44 thf(fact_163_bex__imageD,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P2: A > $o] :
% 0.17/0.44 ( ? [X5: A] :
% 0.17/0.44 ( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A4 ) )
% 0.17/0.44 & ( P2 @ X5 ) )
% 0.17/0.44 => ? [X4: B] :
% 0.17/0.44 ( ( member @ B @ X4 @ A4 )
% 0.17/0.44 & ( P2 @ ( F2 @ X4 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bex_imageD
% 0.17/0.44 thf(fact_164_image__iff,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,Z3: A,F2: B > A,A4: set @ B] :
% 0.17/0.44 ( ( member @ A @ Z3 @ ( image @ B @ A @ F2 @ A4 ) )
% 0.17/0.44 = ( ? [X: B] :
% 0.17/0.44 ( ( member @ B @ X @ A4 )
% 0.17/0.44 & ( Z3
% 0.17/0.44 = ( F2 @ X ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % image_iff
% 0.17/0.44 thf(fact_165_imageI,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,X2: A,A4: set @ A,F2: A > B] :
% 0.17/0.44 ( ( member @ A @ X2 @ A4 )
% 0.17/0.44 => ( member @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % imageI
% 0.17/0.44 thf(fact_166_inj__on__image__iff,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,A4: set @ A,G2: A > B,F2: A > A] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( member @ A @ X4 @ A4 )
% 0.17/0.44 => ! [Xa: A] :
% 0.17/0.44 ( ( member @ A @ Xa @ A4 )
% 0.17/0.44 => ( ( ( G2 @ ( F2 @ X4 ) )
% 0.17/0.44 = ( G2 @ ( F2 @ Xa ) ) )
% 0.17/0.44 = ( ( G2 @ X4 )
% 0.17/0.44 = ( G2 @ Xa ) ) ) ) )
% 0.17/0.44 => ( ( inj_on @ A @ A @ F2 @ A4 )
% 0.17/0.44 => ( ( inj_on @ A @ B @ G2 @ ( image @ A @ A @ F2 @ A4 ) )
% 0.17/0.44 = ( inj_on @ A @ B @ G2 @ A4 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_on_image_iff
% 0.17/0.44 thf(fact_167_surjD,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: B > A,Y: A] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ? [X4: B] :
% 0.17/0.44 ( Y
% 0.17/0.44 = ( F2 @ X4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surjD
% 0.17/0.44 thf(fact_168_surjE,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: B > A,Y: A] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ~ ! [X4: B] :
% 0.17/0.44 ( Y
% 0.17/0.44 != ( F2 @ X4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surjE
% 0.17/0.44 thf(fact_169_surjI,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,G2: B > A,F2: A > B] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( G2 @ ( F2 @ X4 ) )
% 0.17/0.44 = X4 )
% 0.17/0.44 => ( ( image @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surjI
% 0.17/0.44 thf(fact_170_surj__def,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: B > A] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( ! [Y3: A] :
% 0.17/0.44 ? [X: B] :
% 0.17/0.44 ( Y3
% 0.17/0.44 = ( F2 @ X ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surj_def
% 0.17/0.44 thf(fact_171_rangeI,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: B > A,X2: B] : ( member @ A @ ( F2 @ X2 ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % rangeI
% 0.17/0.44 thf(fact_172_range__eqI,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,B2: A,F2: B > A,X2: B] :
% 0.17/0.44 ( ( B2
% 0.17/0.44 = ( F2 @ X2 ) )
% 0.17/0.44 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % range_eqI
% 0.17/0.44 thf(fact_173_inj__on__imp__inj__on__swap,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A2: A,B2: A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( ( member @ A @ A2 @ A4 )
% 0.17/0.44 => ( ( member @ A @ B2 @ A4 )
% 0.17/0.44 => ( inj_on @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_on_imp_inj_on_swap
% 0.17/0.44 thf(fact_174_the__inv__into__f__eq,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: B] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( ( ( F2 @ X2 )
% 0.17/0.44 = Y )
% 0.17/0.44 => ( ( member @ A @ X2 @ A4 )
% 0.17/0.44 => ( ( the_inv_into @ A @ B @ A4 @ F2 @ Y )
% 0.17/0.44 = X2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % the_inv_into_f_eq
% 0.17/0.44 thf(fact_175_the__inv__into__f__f,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( ( member @ A @ X2 @ A4 )
% 0.17/0.44 => ( ( the_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ X2 ) )
% 0.17/0.44 = X2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % the_inv_into_f_f
% 0.17/0.44 thf(fact_176_f__the__inv__into__f__bij__betw,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B,X2: B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.44 => ( ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.44 => ( member @ B @ X2 @ B4 ) )
% 0.17/0.44 => ( ( F2 @ ( the_inv_into @ A @ B @ A4 @ F2 @ X2 ) )
% 0.17/0.44 = X2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % f_the_inv_into_f_bij_betw
% 0.17/0.44 thf(fact_177_bij__betw__the__inv__into,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.44 => ( bij_betw @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ B4 @ A4 ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_betw_the_inv_into
% 0.17/0.44 thf(fact_178_swap__general,axiom,
% 0.17/0.44 ! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
% 0.17/0.44 ( ( A2 != B2 )
% 0.17/0.44 => ( ( C2 != D2 )
% 0.17/0.44 => ( ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
% 0.17/0.44 = ( id @ A ) )
% 0.17/0.44 | ? [X4: A,Y2: A,Z4: A] :
% 0.17/0.44 ( ( X4 != A2 )
% 0.17/0.44 & ( Y2 != A2 )
% 0.17/0.44 & ( Z4 != A2 )
% 0.17/0.44 & ( X4 != Y2 )
% 0.17/0.44 & ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
% 0.17/0.44 = ( comp @ A @ A @ A @ ( swap @ A @ A @ X4 @ Y2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ Z4 @ ( id @ A ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % swap_general
% 0.17/0.44 thf(fact_179_swap__id__common,axiom,
% 0.17/0.44 ! [A: $tType,A2: A,C2: A,B2: A] :
% 0.17/0.44 ( ( A2 != C2 )
% 0.17/0.44 => ( ( B2 != C2 )
% 0.17/0.44 => ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ C2 @ ( id @ A ) ) )
% 0.17/0.44 = ( comp @ A @ A @ A @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % swap_id_common
% 0.17/0.44 thf(fact_180_swap__id__common_H,axiom,
% 0.17/0.44 ! [A: $tType,A2: A,B2: A,C2: A] :
% 0.17/0.44 ( ( A2 != B2 )
% 0.17/0.44 => ( ( A2 != C2 )
% 0.17/0.44 => ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) )
% 0.17/0.44 = ( comp @ A @ A @ A @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % swap_id_common'
% 0.17/0.44 thf(fact_181_swap__id__independent,axiom,
% 0.17/0.44 ! [A: $tType,A2: A,C2: A,D2: A,B2: A] :
% 0.17/0.44 ( ( A2 != C2 )
% 0.17/0.44 => ( ( A2 != D2 )
% 0.17/0.44 => ( ( B2 != C2 )
% 0.17/0.44 => ( ( B2 != D2 )
% 0.17/0.44 => ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
% 0.17/0.44 = ( comp @ A @ A @ A @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % swap_id_independent
% 0.17/0.44 thf(fact_182_fun_Oinj__map__strong,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,D: $tType,X2: D > A,Xa2: D > A,F2: A > B,Fa: A > B] :
% 0.17/0.44 ( ! [Z4: A,Za: A] :
% 0.17/0.44 ( ( member @ A @ Z4 @ ( image @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
% 0.17/0.44 => ( ( member @ A @ Za @ ( image @ D @ A @ Xa2 @ ( top_top @ ( set @ D ) ) ) )
% 0.17/0.44 => ( ( ( F2 @ Z4 )
% 0.17/0.44 = ( Fa @ Za ) )
% 0.17/0.44 => ( Z4 = Za ) ) ) )
% 0.17/0.44 => ( ( ( comp @ A @ B @ D @ F2 @ X2 )
% 0.17/0.44 = ( comp @ A @ B @ D @ Fa @ Xa2 ) )
% 0.17/0.44 => ( X2 = Xa2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % fun.inj_map_strong
% 0.17/0.44 thf(fact_183_fun_Omap__cong0,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,D: $tType,X2: D > A,F2: A > B,G2: A > B] :
% 0.17/0.44 ( ! [Z4: A] :
% 0.17/0.44 ( ( member @ A @ Z4 @ ( image @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
% 0.17/0.44 => ( ( F2 @ Z4 )
% 0.17/0.44 = ( G2 @ Z4 ) ) )
% 0.17/0.44 => ( ( comp @ A @ B @ D @ F2 @ X2 )
% 0.17/0.44 = ( comp @ A @ B @ D @ G2 @ X2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % fun.map_cong0
% 0.17/0.44 thf(fact_184_fun_Omap__cong,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,D: $tType,X2: D > A,Ya: D > A,F2: A > B,G2: A > B] :
% 0.17/0.44 ( ( X2 = Ya )
% 0.17/0.44 => ( ! [Z4: A] :
% 0.17/0.44 ( ( member @ A @ Z4 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 0.17/0.44 => ( ( F2 @ Z4 )
% 0.17/0.44 = ( G2 @ Z4 ) ) )
% 0.17/0.44 => ( ( comp @ A @ B @ D @ F2 @ X2 )
% 0.17/0.44 = ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % fun.map_cong
% 0.17/0.44 thf(fact_185_fun_Oset__map,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,D: $tType,F2: A > B,V: D > A] :
% 0.17/0.44 ( ( image @ D @ B @ ( comp @ A @ B @ D @ F2 @ V ) @ ( top_top @ ( set @ D ) ) )
% 0.17/0.44 = ( image @ A @ B @ F2 @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % fun.set_map
% 0.17/0.44 thf(fact_186_comp__surj,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: A > C] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( ( image @ A @ C @ G2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( top_top @ ( set @ C ) ) )
% 0.17/0.44 => ( ( image @ B @ C @ ( comp @ A @ C @ B @ G2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % comp_surj
% 0.17/0.44 thf(fact_187_inj__image__mem__iff,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A2: A,A4: set @ A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ A4 ) )
% 0.17/0.44 = ( member @ A @ A2 @ A4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_image_mem_iff
% 0.17/0.44 thf(fact_188_inj__image__eq__iff,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( ( image @ A @ B @ F2 @ A4 )
% 0.17/0.44 = ( image @ A @ B @ F2 @ B4 ) )
% 0.17/0.44 = ( A4 = B4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_image_eq_iff
% 0.17/0.44 thf(fact_189_range__ex1__eq,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,B2: B] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( member @ B @ B2 @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
% 0.17/0.44 = ( ? [X: A] :
% 0.17/0.44 ( ( B2
% 0.17/0.44 = ( F2 @ X ) )
% 0.17/0.44 & ! [Y3: A] :
% 0.17/0.44 ( ( B2
% 0.17/0.44 = ( F2 @ Y3 ) )
% 0.17/0.44 => ( Y3 = X ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % range_ex1_eq
% 0.17/0.44 thf(fact_190_surj__id,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( image @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surj_id
% 0.17/0.44 thf(fact_191_comp__inj__on__iff,axiom,
% 0.17/0.44 ! [B: $tType,C: $tType,A: $tType,F2: A > B,A4: set @ A,F3: B > C] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( ( inj_on @ B @ C @ F3 @ ( image @ A @ B @ F2 @ A4 ) )
% 0.17/0.44 = ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F3 @ F2 ) @ A4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % comp_inj_on_iff
% 0.17/0.44 thf(fact_192_inj__on__imageI,axiom,
% 0.17/0.44 ! [B: $tType,C: $tType,A: $tType,G2: C > B,F2: A > C,A4: set @ A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G2 @ F2 ) @ A4 )
% 0.17/0.44 => ( inj_on @ C @ B @ G2 @ ( image @ A @ C @ F2 @ A4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_on_imageI
% 0.17/0.44 thf(fact_193_comp__inj__on,axiom,
% 0.17/0.44 ! [B: $tType,C: $tType,A: $tType,F2: A > B,A4: set @ A,G2: B > C] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( ( inj_on @ B @ C @ G2 @ ( image @ A @ B @ F2 @ A4 ) )
% 0.17/0.44 => ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ A4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % comp_inj_on
% 0.17/0.44 thf(fact_194_bij__betw__imp__surj,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ A4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 => ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( top_top @ ( set @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_betw_imp_surj
% 0.17/0.44 thf(fact_195_bij__is__surj,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: A > B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 => ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( top_top @ ( set @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_is_surj
% 0.17/0.44 thf(fact_196_inj__on__imp__bij__betw,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( bij_betw @ A @ B @ F2 @ A4 @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_on_imp_bij_betw
% 0.17/0.44 thf(fact_197_bij__betw__imageI,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( ( ( image @ A @ B @ F2 @ A4 )
% 0.17/0.44 = B4 )
% 0.17/0.44 => ( bij_betw @ A @ B @ F2 @ A4 @ B4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_betw_imageI
% 0.17/0.44 thf(fact_198_bij__betw__def,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( bij_betw @ A @ B )
% 0.17/0.44 = ( ^ [F: A > B,A5: set @ A,B5: set @ B] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F @ A5 )
% 0.17/0.44 & ( ( image @ A @ B @ F @ A5 )
% 0.17/0.44 = B5 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_betw_def
% 0.17/0.44 thf(fact_199_the__inv__f__f,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,X2: A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X2 ) )
% 0.17/0.44 = X2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % the_inv_f_f
% 0.17/0.44 thf(fact_200_bij__def,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: A > B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 & ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_def
% 0.17/0.44 thf(fact_201_bijI,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: A > B] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( top_top @ ( set @ B ) ) )
% 0.17/0.44 => ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bijI
% 0.17/0.44 thf(fact_202_linear__surj__imp__inj,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( euclid925273238_space @ A )
% 0.17/0.44 => ! [F2: A > A] :
% 0.17/0.44 ( ( real_Vector_linear @ A @ A @ F2 )
% 0.17/0.44 => ( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linear_surj_imp_inj
% 0.17/0.44 thf(fact_203_linear__inj__imp__surj,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( euclid925273238_space @ A )
% 0.17/0.44 => ! [F2: A > A] :
% 0.17/0.44 ( ( real_Vector_linear @ A @ A @ F2 )
% 0.17/0.44 => ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linear_inj_imp_surj
% 0.17/0.44 thf(fact_204_surj__fun__eq,axiom,
% 0.17/0.44 ! [B: $tType,C: $tType,A: $tType,F2: B > A,X7: set @ B,G1: A > C,G22: A > C] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ X7 )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ! [X4: B] :
% 0.17/0.44 ( ( member @ B @ X4 @ X7 )
% 0.17/0.44 => ( ( comp @ A @ C @ B @ G1 @ F2 @ X4 )
% 0.17/0.44 = ( comp @ A @ C @ B @ G22 @ F2 @ X4 ) ) )
% 0.17/0.44 => ( G1 = G22 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surj_fun_eq
% 0.17/0.44 thf(fact_205_Inf_OINF__id__eq,axiom,
% 0.17/0.44 ! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
% 0.17/0.44 ( ( Inf @ ( image @ A @ A @ ( id @ A ) @ A4 ) )
% 0.17/0.44 = ( Inf @ A4 ) ) ).
% 0.17/0.44
% 0.17/0.44 % Inf.INF_id_eq
% 0.17/0.44 thf(fact_206_Sup_OSUP__id__eq,axiom,
% 0.17/0.44 ! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
% 0.17/0.44 ( ( Sup @ ( image @ A @ A @ ( id @ A ) @ A4 ) )
% 0.17/0.44 = ( Sup @ A4 ) ) ).
% 0.17/0.44
% 0.17/0.44 % Sup.SUP_id_eq
% 0.17/0.44 thf(fact_207_Inf_OINF__image,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G2: B > A,F2: C > B,A4: set @ C] :
% 0.17/0.44 ( ( Inf @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F2 @ A4 ) ) )
% 0.17/0.44 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ A4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Inf.INF_image
% 0.17/0.44 thf(fact_208_bij__betwI_H,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,X7: set @ A,F2: A > B,Y5: set @ B] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( member @ A @ X4 @ X7 )
% 0.17/0.44 => ! [Y2: A] :
% 0.17/0.44 ( ( member @ A @ Y2 @ X7 )
% 0.17/0.44 => ( ( ( F2 @ X4 )
% 0.17/0.44 = ( F2 @ Y2 ) )
% 0.17/0.44 = ( X4 = Y2 ) ) ) )
% 0.17/0.44 => ( ! [X4: A] :
% 0.17/0.44 ( ( member @ A @ X4 @ X7 )
% 0.17/0.44 => ( member @ B @ ( F2 @ X4 ) @ Y5 ) )
% 0.17/0.44 => ( ! [Y2: B] :
% 0.17/0.44 ( ( member @ B @ Y2 @ Y5 )
% 0.17/0.44 => ? [X5: A] :
% 0.17/0.44 ( ( member @ A @ X5 @ X7 )
% 0.17/0.44 & ( Y2
% 0.17/0.44 = ( F2 @ X5 ) ) ) )
% 0.17/0.44 => ( bij_betw @ A @ B @ F2 @ X7 @ Y5 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_betwI'
% 0.17/0.44 thf(fact_209_Sup_OSUP__image,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G2: B > A,F2: C > B,A4: set @ C] :
% 0.17/0.44 ( ( Sup @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F2 @ A4 ) ) )
% 0.17/0.44 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ A4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % Sup.SUP_image
% 0.17/0.44 thf(fact_210_inj__adjoint__iff__surj,axiom,
% 0.17/0.44 ! [M2: $tType,N2: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ N2 )
% 0.17/0.44 & ( euclid925273238_space @ M2 ) )
% 0.17/0.44 => ! [F2: M2 > N2] :
% 0.17/0.44 ( ( real_Vector_linear @ M2 @ N2 @ F2 )
% 0.17/0.44 => ( ( inj_on @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
% 0.17/0.44 = ( ( image @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) )
% 0.17/0.44 = ( top_top @ ( set @ N2 ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_adjoint_iff_surj
% 0.17/0.44 thf(fact_211_surj__adjoint__iff__inj,axiom,
% 0.17/0.44 ! [N2: $tType,M2: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ M2 )
% 0.17/0.44 & ( euclid925273238_space @ N2 ) )
% 0.17/0.44 => ! [F2: M2 > N2] :
% 0.17/0.44 ( ( real_Vector_linear @ M2 @ N2 @ F2 )
% 0.17/0.44 => ( ( ( image @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
% 0.17/0.44 = ( top_top @ ( set @ M2 ) ) )
% 0.17/0.44 = ( inj_on @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surj_adjoint_iff_inj
% 0.17/0.44 thf(fact_212_adjoint__adjoint,axiom,
% 0.17/0.44 ! [M2: $tType,N2: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ N2 )
% 0.17/0.44 & ( euclid925273238_space @ M2 ) )
% 0.17/0.44 => ! [F2: N2 > M2] :
% 0.17/0.44 ( ( real_Vector_linear @ N2 @ M2 @ F2 )
% 0.17/0.44 => ( ( linear_adjoint @ M2 @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 ) )
% 0.17/0.44 = F2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % adjoint_adjoint
% 0.17/0.44 thf(fact_213_adjoint__linear,axiom,
% 0.17/0.44 ! [M2: $tType,N2: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ N2 )
% 0.17/0.44 & ( euclid925273238_space @ M2 ) )
% 0.17/0.44 => ! [F2: N2 > M2] :
% 0.17/0.44 ( ( real_Vector_linear @ N2 @ M2 @ F2 )
% 0.17/0.44 => ( real_Vector_linear @ M2 @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % adjoint_linear
% 0.17/0.44 thf(fact_214_linear__surj__adj__imp__inj,axiom,
% 0.17/0.44 ! [N2: $tType,M2: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ M2 )
% 0.17/0.44 & ( euclid925273238_space @ N2 ) )
% 0.17/0.44 => ! [F2: M2 > N2] :
% 0.17/0.44 ( ( real_Vector_linear @ M2 @ N2 @ F2 )
% 0.17/0.44 => ( ( ( image @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
% 0.17/0.44 = ( top_top @ ( set @ M2 ) ) )
% 0.17/0.44 => ( inj_on @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linear_surj_adj_imp_inj
% 0.17/0.44 thf(fact_215_coplanar__linear__image__eq,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ A )
% 0.17/0.44 & ( euclid925273238_space @ B ) )
% 0.17/0.44 => ! [F2: A > B,S: set @ A] :
% 0.17/0.44 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.44 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( coplanar @ B @ ( image @ A @ B @ F2 @ S ) )
% 0.17/0.44 = ( coplanar @ A @ S ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % coplanar_linear_image_eq
% 0.17/0.44 thf(fact_216_linear__singular__into__hyperplane,axiom,
% 0.17/0.44 ! [N2: $tType] :
% 0.17/0.44 ( ( euclid925273238_space @ N2 )
% 0.17/0.44 => ! [F2: N2 > N2] :
% 0.17/0.44 ( ( real_Vector_linear @ N2 @ N2 @ F2 )
% 0.17/0.44 => ( ( ~ ( inj_on @ N2 @ N2 @ F2 @ ( top_top @ ( set @ N2 ) ) ) )
% 0.17/0.44 = ( ? [A7: N2] :
% 0.17/0.44 ( ( A7
% 0.17/0.44 != ( zero_zero @ N2 ) )
% 0.17/0.44 & ! [X: N2] :
% 0.17/0.44 ( ( inner_780170721_inner @ N2 @ A7 @ ( F2 @ X ) )
% 0.17/0.44 = ( zero_zero @ real ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linear_singular_into_hyperplane
% 0.17/0.44 thf(fact_217_adjoint__unique,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType] :
% 0.17/0.44 ( ( ( inner_real_inner @ B )
% 0.17/0.44 & ( inner_real_inner @ A ) )
% 0.17/0.44 => ! [F2: A > B,G2: B > A] :
% 0.17/0.44 ( ! [X4: A,Y2: B] :
% 0.17/0.44 ( ( inner_780170721_inner @ B @ ( F2 @ X4 ) @ Y2 )
% 0.17/0.44 = ( inner_780170721_inner @ A @ X4 @ ( G2 @ Y2 ) ) )
% 0.17/0.44 => ( ( linear_adjoint @ A @ B @ F2 )
% 0.17/0.44 = G2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % adjoint_unique
% 0.17/0.44 thf(fact_218_hyperplane__eq__Ex,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( inner_real_inner @ A )
% 0.17/0.44 => ! [A2: A,B2: real] :
% 0.17/0.44 ( ( A2
% 0.17/0.44 != ( zero_zero @ A ) )
% 0.17/0.44 => ~ ! [X4: A] :
% 0.17/0.44 ( ( inner_780170721_inner @ A @ A2 @ X4 )
% 0.17/0.44 != B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % hyperplane_eq_Ex
% 0.17/0.44 thf(fact_219_adjoint__works,axiom,
% 0.17/0.44 ! [N2: $tType,M2: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ M2 )
% 0.17/0.44 & ( euclid925273238_space @ N2 ) )
% 0.17/0.44 => ! [F2: N2 > M2,X2: N2,Y: M2] :
% 0.17/0.44 ( ( real_Vector_linear @ N2 @ M2 @ F2 )
% 0.17/0.44 => ( ( inner_780170721_inner @ N2 @ X2 @ ( linear_adjoint @ N2 @ M2 @ F2 @ Y ) )
% 0.17/0.44 = ( inner_780170721_inner @ M2 @ ( F2 @ X2 ) @ Y ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % adjoint_works
% 0.17/0.44 thf(fact_220_adjoint__clauses_I2_J,axiom,
% 0.17/0.44 ! [M2: $tType,N2: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ N2 )
% 0.17/0.44 & ( euclid925273238_space @ M2 ) )
% 0.17/0.44 => ! [F2: N2 > M2,Y: M2,X2: N2] :
% 0.17/0.44 ( ( real_Vector_linear @ N2 @ M2 @ F2 )
% 0.17/0.44 => ( ( inner_780170721_inner @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 @ Y ) @ X2 )
% 0.17/0.44 = ( inner_780170721_inner @ M2 @ Y @ ( F2 @ X2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % adjoint_clauses(2)
% 0.17/0.44 thf(fact_221_vector__eq,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( inner_real_inner @ A )
% 0.17/0.44 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.17/0.44 = ( ^ [X: A,Y3: A] :
% 0.17/0.44 ( ( ( inner_780170721_inner @ A @ X @ X )
% 0.17/0.44 = ( inner_780170721_inner @ A @ X @ Y3 ) )
% 0.17/0.44 & ( ( inner_780170721_inner @ A @ Y3 @ Y3 )
% 0.17/0.44 = ( inner_780170721_inner @ A @ X @ X ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % vector_eq
% 0.17/0.44 thf(fact_222_vector__eq__ldot,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( inner_real_inner @ A )
% 0.17/0.44 => ! [Y: A,Z3: A] :
% 0.17/0.44 ( ( ! [X: A] :
% 0.17/0.44 ( ( inner_780170721_inner @ A @ X @ Y )
% 0.17/0.44 = ( inner_780170721_inner @ A @ X @ Z3 ) ) )
% 0.17/0.44 = ( Y = Z3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % vector_eq_ldot
% 0.17/0.44 thf(fact_223_vector__eq__rdot,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( inner_real_inner @ A )
% 0.17/0.44 => ! [X2: A,Y: A] :
% 0.17/0.44 ( ( ! [Z2: A] :
% 0.17/0.44 ( ( inner_780170721_inner @ A @ X2 @ Z2 )
% 0.17/0.44 = ( inner_780170721_inner @ A @ Y @ Z2 ) ) )
% 0.17/0.44 = ( X2 = Y ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % vector_eq_rdot
% 0.17/0.44 thf(fact_224_coplanar__linear__image,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ A )
% 0.17/0.44 & ( real_V55928688vector @ B ) )
% 0.17/0.44 => ! [S: set @ A,F2: A > B] :
% 0.17/0.44 ( ( coplanar @ A @ S )
% 0.17/0.44 => ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.44 => ( coplanar @ B @ ( image @ A @ B @ F2 @ S ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % coplanar_linear_image
% 0.17/0.44 thf(fact_225_inner__eq__zero__iff,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( inner_real_inner @ A )
% 0.17/0.44 => ! [X2: A] :
% 0.17/0.44 ( ( ( inner_780170721_inner @ A @ X2 @ X2 )
% 0.17/0.44 = ( zero_zero @ real ) )
% 0.17/0.44 = ( X2
% 0.17/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inner_eq_zero_iff
% 0.17/0.44 thf(fact_226_inner__zero__right,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( inner_real_inner @ A )
% 0.17/0.44 => ! [X2: A] :
% 0.17/0.44 ( ( inner_780170721_inner @ A @ X2 @ ( zero_zero @ A ) )
% 0.17/0.44 = ( zero_zero @ real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inner_zero_right
% 0.17/0.44 thf(fact_227_all__zero__iff,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( inner_real_inner @ A )
% 0.17/0.44 => ! [X2: A] :
% 0.17/0.44 ( ( ! [U: A] :
% 0.17/0.44 ( ( inner_780170721_inner @ A @ X2 @ U )
% 0.17/0.44 = ( zero_zero @ real ) ) )
% 0.17/0.44 = ( X2
% 0.17/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % all_zero_iff
% 0.17/0.44 thf(fact_228_inner__zero__left,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( inner_real_inner @ A )
% 0.17/0.44 => ! [X2: A] :
% 0.17/0.44 ( ( inner_780170721_inner @ A @ ( zero_zero @ A ) @ X2 )
% 0.17/0.44 = ( zero_zero @ real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inner_zero_left
% 0.17/0.44 thf(fact_229_aff__dim__injective__linear__image,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType] :
% 0.17/0.44 ( ( ( euclid925273238_space @ A )
% 0.17/0.44 & ( euclid925273238_space @ B ) )
% 0.17/0.44 => ! [F2: A > B,S: set @ A] :
% 0.17/0.44 ( ( real_Vector_linear @ A @ B @ F2 )
% 0.17/0.44 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( aff_dim @ B @ ( image @ A @ B @ F2 @ S ) )
% 0.17/0.44 = ( aff_dim @ A @ S ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % aff_dim_injective_linear_image
% 0.17/0.44 thf(fact_230_bij__swap__comp,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,P: A > B,A2: B,B2: B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 => ( ( comp @ B @ B @ A @ ( swap @ B @ B @ A2 @ B2 @ ( id @ B ) ) @ P )
% 0.17/0.44 = ( swap @ A @ B @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ A2 ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ B2 ) @ P ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_swap_comp
% 0.17/0.44 thf(fact_231_inj__linear__imp__inv__bounded__linear,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( euclid925273238_space @ A )
% 0.17/0.44 => ! [F2: A > A] :
% 0.17/0.44 ( ( real_V1632203528linear @ A @ A @ F2 )
% 0.17/0.44 => ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( real_V1632203528linear @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_linear_imp_inv_bounded_linear
% 0.17/0.44 thf(fact_232_inv__swap__id,axiom,
% 0.17/0.44 ! [A: $tType,A2: A,B2: A] :
% 0.17/0.44 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) )
% 0.17/0.44 = ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_swap_id
% 0.17/0.44 thf(fact_233_inj__linear__imp__inv__linear,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( euclid925273238_space @ A )
% 0.17/0.44 => ! [F2: A > A] :
% 0.17/0.44 ( ( real_Vector_linear @ A @ A @ F2 )
% 0.17/0.44 => ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( real_Vector_linear @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_linear_imp_inv_linear
% 0.17/0.44 thf(fact_234_inv__o__cancel,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
% 0.17/0.44 = ( id @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_o_cancel
% 0.17/0.44 thf(fact_235_o__inv__o__cancel,axiom,
% 0.17/0.44 ! [B: $tType,C: $tType,A: $tType,F2: A > B,G2: A > C] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G2 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ F2 )
% 0.17/0.44 = G2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % o_inv_o_cancel
% 0.17/0.44 thf(fact_236_inv__into__f__f,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( ( member @ A @ X2 @ A4 )
% 0.17/0.44 => ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ X2 ) )
% 0.17/0.44 = X2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_into_f_f
% 0.17/0.44 thf(fact_237_inv__id,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
% 0.17/0.44 = ( id @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_id
% 0.17/0.44 thf(fact_238_inv__equality,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,G2: B > A,F2: A > B] :
% 0.17/0.44 ( ! [X4: A] :
% 0.17/0.44 ( ( G2 @ ( F2 @ X4 ) )
% 0.17/0.44 = X4 )
% 0.17/0.44 => ( ! [Y2: B] :
% 0.17/0.44 ( ( F2 @ ( G2 @ Y2 ) )
% 0.17/0.44 = Y2 )
% 0.17/0.44 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
% 0.17/0.44 = G2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_equality
% 0.17/0.44 thf(fact_239_inv__into__f__eq,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: B] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.17/0.44 => ( ( member @ A @ X2 @ A4 )
% 0.17/0.44 => ( ( ( F2 @ X2 )
% 0.17/0.44 = Y )
% 0.17/0.44 => ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ Y )
% 0.17/0.44 = X2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_into_f_eq
% 0.17/0.44 thf(fact_240_bij__betw__inv__into,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 0.17/0.44 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ B4 @ A4 ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_betw_inv_into
% 0.17/0.44 thf(fact_241_inv__into__inv__into__eq,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A8: set @ B,A2: A] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 0.17/0.44 => ( ( member @ A @ A2 @ A4 )
% 0.17/0.44 => ( ( hilbert_inv_into @ B @ A @ A8 @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ A2 )
% 0.17/0.44 = ( F2 @ A2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_into_inv_into_eq
% 0.17/0.44 thf(fact_242_bij__betw__inv__into__left,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A8: set @ B,A2: A] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 0.17/0.44 => ( ( member @ A @ A2 @ A4 )
% 0.17/0.44 => ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ A2 ) )
% 0.17/0.44 = A2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_betw_inv_into_left
% 0.17/0.44 thf(fact_243_bij__betw__inv__into__right,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,A8: set @ B,A10: B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 0.17/0.44 => ( ( member @ B @ A10 @ A8 )
% 0.17/0.44 => ( ( F2 @ ( hilbert_inv_into @ A @ B @ A4 @ F2 @ A10 ) )
% 0.17/0.44 = A10 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_betw_inv_into_right
% 0.17/0.44 thf(fact_244_surj__f__inv__f,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: B > A,Y: A] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ Y ) )
% 0.17/0.44 = Y ) ) ).
% 0.17/0.44
% 0.17/0.44 % surj_f_inv_f
% 0.17/0.44 thf(fact_245_surj__iff__all,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: B > A] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 = ( ! [X: A] :
% 0.17/0.44 ( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ X ) )
% 0.17/0.44 = X ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surj_iff_all
% 0.17/0.44 thf(fact_246_image__f__inv__f,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: B > A,A4: set @ A] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( image @ B @ A @ F2 @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ A4 ) )
% 0.17/0.44 = A4 ) ) ).
% 0.17/0.44
% 0.17/0.44 % image_f_inv_f
% 0.17/0.44 thf(fact_247_surj__imp__inv__eq,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: B > A,G2: A > B] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ! [X4: B] :
% 0.17/0.44 ( ( G2 @ ( F2 @ X4 ) )
% 0.17/0.44 = X4 )
% 0.17/0.44 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
% 0.17/0.44 = G2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surj_imp_inv_eq
% 0.17/0.44 thf(fact_248_inv__f__f,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,X2: A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X2 ) )
% 0.17/0.44 = X2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_f_f
% 0.17/0.44 thf(fact_249_inv__f__eq,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B,X2: A,Y: B] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ( ( F2 @ X2 )
% 0.17/0.44 = Y )
% 0.17/0.44 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y )
% 0.17/0.44 = X2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_f_eq
% 0.17/0.44 thf(fact_250_inj__imp__inv__eq,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,F2: A > B,G2: B > A] :
% 0.17/0.44 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( ! [X4: B] :
% 0.17/0.44 ( ( F2 @ ( G2 @ X4 ) )
% 0.17/0.44 = X4 )
% 0.17/0.44 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
% 0.17/0.44 = G2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % inj_imp_inv_eq
% 0.17/0.44 thf(fact_251_bij__imp__bij__inv,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_imp_bij_inv
% 0.17/0.44 thf(fact_252_bij__inv__eq__iff,axiom,
% 0.17/0.44 ! [A: $tType,B: $tType,P: A > B,X2: A,Y: B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 => ( ( X2
% 0.17/0.44 = ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ Y ) )
% 0.17/0.44 = ( ( P @ X2 )
% 0.17/0.44 = Y ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % bij_inv_eq_iff
% 0.17/0.44 thf(fact_253_inv__inv__eq,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: A > B] :
% 0.17/0.44 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) )
% 0.17/0.44 = F2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % inv_inv_eq
% 0.17/0.44 thf(fact_254_surj__imp__inj__inv,axiom,
% 0.17/0.44 ! [B: $tType,A: $tType,F2: B > A] :
% 0.17/0.44 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.44 = ( top_top @ ( set @ A ) ) )
% 0.17/0.44 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % surj_imp_inj_inv
% 0.17/0.44
% 0.17/0.44 % Subclasses (7)
% 0.17/0.44 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___HOL_Otype,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( execut510477386_space @ A )
% 0.17/0.44 => ( type @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( execut510477386_space @ A )
% 0.17/0.44 => ( zero @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Inner__Product_Oreal__inner,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( execut510477386_space @ A )
% 0.17/0.44 => ( inner_real_inner @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Oreal__vector,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( execut510477386_space @ A )
% 0.17/0.44 => ( real_V1076094709vector @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Euclidean__Space_Oeuclidean__space,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( execut510477386_space @ A )
% 0.17/0.44 => ( euclid925273238_space @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Topological__Spaces_Otopological__space,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( execut510477386_space @ A )
% 0.17/0.44 => ( topolo503727757_space @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 0.17/0.44 ! [A: $tType] :
% 0.17/0.44 ( ( execut510477386_space @ A )
% 0.17/0.44 => ( real_V55928688vector @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % Type constructors (21)
% 0.17/0.44 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 0.17/0.44 ! [A11: $tType,A12: $tType] :
% 0.17/0.44 ( ( topolo503727757_space @ A12 )
% 0.17/0.44 => ( topolo503727757_space @ ( A11 > A12 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_fun___Orderings_Otop,axiom,
% 0.17/0.44 ! [A11: $tType,A12: $tType] :
% 0.17/0.44 ( ( top @ A12 )
% 0.17/0.44 => ( top @ ( A11 > A12 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_1,axiom,
% 0.17/0.44 topolo503727757_space @ int ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.17/0.44 zero @ int ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
% 0.17/0.44 ! [A11: $tType] : ( top @ ( set @ A11 ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Set_Oset___Groups_Ozero_3,axiom,
% 0.17/0.44 ! [A11: $tType] :
% 0.17/0.44 ( ( zero @ A11 )
% 0.17/0.44 => ( zero @ ( set @ A11 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_4,axiom,
% 0.17/0.44 topolo503727757_space @ $o ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_HOL_Obool___Orderings_Otop_5,axiom,
% 0.17/0.44 top @ $o ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
% 0.17/0.44 execut510477386_space @ real ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 0.17/0.44 real_V55928688vector @ real ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_6,axiom,
% 0.17/0.44 topolo503727757_space @ real ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
% 0.17/0.44 euclid925273238_space @ real ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
% 0.17/0.44 real_V1076094709vector @ real ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Real_Oreal___Inner__Product_Oreal__inner,axiom,
% 0.17/0.44 inner_real_inner @ real ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Real_Oreal___Groups_Ozero_7,axiom,
% 0.17/0.44 zero @ real ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_8,axiom,
% 0.17/0.44 real_V55928688vector @ complex ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_9,axiom,
% 0.17/0.44 topolo503727757_space @ complex ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Complex_Ocomplex___Euclidean__Space_Oeuclidean__space_10,axiom,
% 0.17/0.44 euclid925273238_space @ complex ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_11,axiom,
% 0.17/0.44 real_V1076094709vector @ complex ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Complex_Ocomplex___Inner__Product_Oreal__inner_12,axiom,
% 0.17/0.44 inner_real_inner @ complex ).
% 0.17/0.44
% 0.17/0.44 thf(tcon_Complex_Ocomplex___Groups_Ozero_13,axiom,
% 0.17/0.44 zero @ complex ).
% 0.17/0.44
% 0.17/0.44 % Free types (1)
% 0.17/0.44 thf(tfree_0,hypothesis,
% 0.17/0.44 execut510477386_space @ a ).
% 0.17/0.44
% 0.17/0.44 % Conjectures (1)
% 0.17/0.44 thf(conj_0,conjecture,
% 0.17/0.44 ( ( path_pathfinish @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) )
% 0.17/0.44 = ( path_pathstart @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) ) ) ).
% 0.17/0.44
% 0.17/0.44 %------------------------------------------------------------------------------
% 0.17/0.44 ------------------------
% 3.76/1.96 ---- Embedded in THF ---
% 3.76/1.96 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.76/1.96 %%% Generated on Tue Feb 24 08:27:46 EST 2026
% 3.76/1.96 %%% using '$$dhol' embedding, version 1.3.0.
% 3.76/1.96 %%% Logic specification used:
% 3.76/1.96 %%% thf(spec, logic, $$dhol).
% 3.76/1.96
% 3.76/1.96 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2DTF11658.p
% 3.76/1.96 thf(ty_t_Bounded__Linear__Function_Oblinfun, type, bounde2145540817linfun: ($tType > ($tType > $tType))).
% 3.76/1.96 thf(ty_t_Complex_Ocomplex, type, complex: $tType).
% 3.76/1.96 thf(ty_t_Real_Oreal, type, real: $tType).
% 3.76/1.96 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.76/1.96 thf(ty_t_Int_Oint, type, int: $tType).
% 3.76/1.96 thf(ty_tf_a, type, a: $tType).
% 3.76/1.96 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_cl_Inner__Product_Oreal__inner, type, inner_real_inner: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_cl_Real__Vector__Spaces_Oreal__vector, type, real_V1076094709vector: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_cl_Euclidean__Space_Oeuclidean__space, type, euclid925273238_space: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_cl_Topological__Spaces_Otopological__space, type, topolo503727757_space: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector, type, real_V55928688vector: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space, type, execut510477386_space: (!> [A:$tType]: ($o))).
% 3.76/1.96 thf(sy_c_Affine_Oaff__dim, type, aff_dim: (!> [A:$tType]: (((set @ A) > int)))).
% 3.76/1.96 thf(sy_c_Fun_Obij__betw, type, bij_betw: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > ((set @ B) > $o)))))).
% 3.76/1.96 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 3.76/1.96 thf(sy_c_Fun_Oid, type, id: (!> [A:$tType]: ((A > A)))).
% 3.76/1.96 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 3.76/1.96 thf(sy_c_Fun_Oswap, type, swap: (!> [A:$tType,B:$tType]: ((A > (A > ((A > B) > (A > B))))))).
% 3.76/1.96 thf(sy_c_Fun_Othe__inv__into, type, the_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 3.76/1.96 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 3.76/1.96 thf(sy_c_Hilbert__Choice_Oinv__into, type, hilbert_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 3.76/1.96 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner, type, inner_780170721_inner: (!> [A:$tType]: ((A > (A > real))))).
% 3.76/1.96 thf(sy_c_Linear__Algebra_Oadjoint, type, linear_adjoint: (!> [A:$tType,B:$tType]: (((A > B) > (B > A))))).
% 3.76/1.96 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 3.76/1.96 thf(sy_c_Path__Connected_Oarc, type, path_arc: (!> [A:$tType]: (((real > A) > $o)))).
% 3.76/1.96 thf(sy_c_Path__Connected_Opath, type, path_path: (!> [A:$tType]: (((real > A) > $o)))).
% 3.76/1.96 thf(sy_c_Path__Connected_Opathfinish, type, path_pathfinish: (!> [A:$tType]: (((real > A) > A)))).
% 3.76/1.96 thf(sy_c_Path__Connected_Opathstart, type, path_pathstart: (!> [A:$tType]: (((real > A) > A)))).
% 3.76/1.96 thf(sy_c_Path__Connected_Orectpath, type, path_rectpath: (complex > (complex > (real > complex)))).
% 3.76/1.96 thf(sy_c_Path__Connected_Osimple__path, type, path_simple_path: (!> [A:$tType]: (((real > A) > $o)))).
% 3.76/1.96 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2, type, poinca1076805489pen_R2: (!> [A:$tType]: (((A > A) > ((A > ((bounde2145540817linfun @ A) @ A)) > ((set @ A) > $o)))))).
% 3.76/1.96 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Ocomplex__of, type, poinca1531302022lex_of: (!> [A:$tType]: ((A > complex)))).
% 3.76/1.96 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Oreal__of, type, poinca179318102eal_of: (!> [A:$tType]: ((complex > A)))).
% 3.76/1.96 thf(sy_c_Real__Vector__Spaces_Obounded__linear, type, real_V1632203528linear: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 3.76/1.96 thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms, type, real_V1072672427axioms: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 3.76/1.96 thf(sy_c_Real__Vector__Spaces_Olinear, type, real_Vector_linear: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 3.76/1.96 thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist, type, iso_tu2011167877assist: (!> [B:$tType,A:$tType]: ((((B > B) > (A > A)) > ((A > B) > (A > ((B > B) > (A > (B > $o))))))))).
% 3.76/1.96 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.76/1.96 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.76/1.96 thf(sy_c_Starlike_Ocoplanar, type, coplanar: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.76/1.96 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.76/1.96 thf(sy_v_c, type, c: (real > a)).
% 3.76/1.96 thf(fact_0_assms_I2_J, axiom, (((path_pathfinish @ a) @ c) = ((path_pathstart @ a) @ c))).
% 3.76/1.96 thf(fact_1_assms_I1_J, axiom, ((path_simple_path @ a) @ c)).
% 3.76/1.96 thf(fact_2_a1, axiom, ((path_simple_path @ complex) @ (((((comp @ a) @ complex) @ real) @ (poinca1531302022lex_of @ a)) @ c))).
% 3.76/1.96 thf(fact_3_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F:(B > A),G:(C > B),X:C]: ((F @ (G @ X)))))))).
% 3.76/1.96 thf(fact_4_pathfinish__compose, axiom, (! [A:$tType,B:$tType]: ((((topolo503727757_space @ B) & (topolo503727757_space @ A)) => (! [F2:(B > A),P:(real > B)]: ((((path_pathfinish @ A) @ (((((comp @ B) @ A) @ real) @ F2) @ P)) = (F2 @ ((path_pathfinish @ B) @ P))))))))).
% 3.76/1.96 thf(fact_5_pathstart__compose, axiom, (! [A:$tType,B:$tType]: ((((topolo503727757_space @ B) & (topolo503727757_space @ A)) => (! [F2:(B > A),P:(real > B)]: ((((path_pathstart @ A) @ (((((comp @ B) @ A) @ real) @ F2) @ P)) = (F2 @ ((path_pathstart @ B) @ P))))))))).
% 3.76/1.96 thf(fact_6_complex__of__bounded__linear, axiom, (((real_V1632203528linear @ a) @ complex) @ (poinca1531302022lex_of @ a))).
% 3.76/1.96 thf(fact_7_complex__of__linear, axiom, (((real_Vector_linear @ a) @ complex) @ (poinca1531302022lex_of @ a))).
% 3.76/1.96 thf(fact_8_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F:(B > C),G:(A > B),X:A]: ((F @ (G @ X)))))))).
% 3.76/1.96 thf(fact_9_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F2:(D > B),G2:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F2) @ G2)) @ H) = (((((comp @ D) @ B) @ A) @ F2) @ (((((comp @ C) @ D) @ A) @ G2) @ H)))))).
% 3.76/1.96 thf(fact_10_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(D > B),D2:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => ((A2 @ (B2 @ V)) = (C2 @ (D2 @ V))))))).
% 3.76/1.96 thf(fact_11_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(D > B),D2:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => (! [V2:A]: (((A2 @ (B2 @ V2)) = (C2 @ (D2 @ V2))))))))).
% 3.76/1.96 thf(fact_12_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F2:(B > A),G2:(C > B),X2:C,F3:(D > A),G3:(E > D),X3:E]: ((((F2 @ (G2 @ X2)) = (F3 @ (G3 @ X3))) => (((((((comp @ B) @ A) @ C) @ F2) @ G2) @ X2) = ((((((comp @ D) @ A) @ E) @ F3) @ G3) @ X3)))))).
% 3.76/1.96 thf(fact_13_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = C2) => ((A2 @ (B2 @ V)) = (C2 @ V)))))).
% 3.76/1.96 thf(fact_14_pathstart__linear__image__eq, axiom, (! [B:$tType,A:$tType]: ((((real_V1076094709vector @ A) & ((topolo503727757_space @ A) & ((real_V1076094709vector @ B) & (topolo503727757_space @ B)))) => (! [F2:(A > B),G2:(real > A)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((path_pathstart @ B) @ (((((comp @ A) @ B) @ real) @ F2) @ G2)) = (F2 @ ((path_pathstart @ A) @ G2)))))))))).
% 3.76/1.96 thf(fact_15_pathfinish__linear__image, axiom, (! [B:$tType,A:$tType]: ((((real_V1076094709vector @ A) & ((topolo503727757_space @ A) & ((real_V1076094709vector @ B) & (topolo503727757_space @ B)))) => (! [F2:(A > B),G2:(real > A)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((path_pathfinish @ B) @ (((((comp @ A) @ B) @ real) @ F2) @ G2)) = (F2 @ ((path_pathfinish @ A) @ G2)))))))))).
% 3.76/1.96 thf(fact_16_bounded__linear_Olinear, axiom, (! [B:$tType,A:$tType]: ((((real_V55928688vector @ A) & (real_V55928688vector @ B)) => (! [F2:(A > B)]: (((((real_V1632203528linear @ A) @ B) @ F2) => (((real_Vector_linear @ A) @ B) @ F2)))))))).
% 3.76/1.96 thf(fact_17_linear__conv__bounded__linear, axiom, (! [B:$tType,A:$tType]: ((((euclid925273238_space @ A) & (real_V55928688vector @ B)) => (((real_Vector_linear @ A) @ B) = ((real_V1632203528linear @ A) @ B)))))).
% 3.76/1.96 thf(fact_18_linear__compose, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((real_V1076094709vector @ A) & ((real_V1076094709vector @ B) & (real_V1076094709vector @ C))) => (! [F2:(A > B),G2:(B > C)]: (((((real_Vector_linear @ A) @ B) @ F2) => ((((real_Vector_linear @ B) @ C) @ G2) => (((real_Vector_linear @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G2) @ F2)))))))))).
% 3.76/1.96 thf(fact_19__092_060open_062_092_060And_062g_O_Ainj_Acomplex__of_A_092_060Longrightarrow_062_Asimple__path_A_Icomplex__of_A_092_060circ_062_Ag_J_A_061_Asimple__path_Ag_092_060close_062, axiom, (! [G2:(real > a)]: ((((((inj_on @ a) @ complex) @ (poinca1531302022lex_of @ a)) @ (top_top @ (set @ a))) => (((path_simple_path @ complex) @ (((((comp @ a) @ complex) @ real) @ (poinca1531302022lex_of @ a)) @ G2)) = ((path_simple_path @ a) @ G2)))))).
% 3.76/1.96 thf(fact_20_pathfinish__rectpath, axiom, (! [A1:complex,A3:complex]: ((((path_pathfinish @ complex) @ ((path_rectpath @ A1) @ A3)) = A1)))).
% 3.76/1.96 thf(fact_21_pathstart__rectpath, axiom, (! [A1:complex,A3:complex]: ((((path_pathstart @ complex) @ ((path_rectpath @ A1) @ A3)) = A1)))).
% 3.76/1.96 thf(fact_22_bounded__linear_Obounded__linear, axiom, (! [B:$tType,A:$tType]: ((((real_V55928688vector @ A) & (real_V55928688vector @ B)) => (! [F2:(A > B)]: (((((real_V1632203528linear @ A) @ B) @ F2) => (((real_V1632203528linear @ A) @ B) @ F2)))))))).
% 3.76/1.96 thf(fact_23_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G2:(C > B),H:(A > C),R1:(D > B),R2:(A > D),F2:(B > E),L:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G2) @ H) = (((((comp @ D) @ B) @ A) @ R1) @ R2)) => (((((((comp @ B) @ E) @ D) @ F2) @ R1) = L) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F2) @ G2)) @ H) = (((((comp @ D) @ E) @ A) @ L) @ R2))))))).
% 3.76/1.96 thf(fact_24_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F2:(C > B),G2:(A > C),L1:(D > B),L2:(A > D),H:(E > A),R:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F2) @ G2) = (((((comp @ D) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D) @ E) @ L2) @ H) = R) => ((((((comp @ C) @ B) @ E) @ F2) @ (((((comp @ A) @ C) @ E) @ G2) @ H)) = (((((comp @ D) @ B) @ E) @ L1) @ R))))))).
% 3.76/1.96 thf(fact_25_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G2:(C > B),H:(A > C),R:(A > B),F2:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G2) @ H) = R) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F2) @ G2)) @ H) = (((((comp @ B) @ D) @ A) @ F2) @ R)))))).
% 3.76/1.96 thf(fact_26_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F2:(C > B),G2:(A > C),L:(A > B),H:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F2) @ G2) = L) => ((((((comp @ C) @ B) @ D) @ F2) @ (((((comp @ A) @ C) @ D) @ G2) @ H)) = (((((comp @ A) @ B) @ D) @ L) @ H)))))).
% 3.76/1.96 thf(fact_27_injD, axiom, (! [B:$tType,A:$tType,F2:(A > B),X2:A,Y:A]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((F2 @ X2) = (F2 @ Y)) => (X2 = Y)))))).
% 3.76/1.96 thf(fact_28_injI, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: (((! [X4:A,Y2:A]: ((((F2 @ X4) = (F2 @ Y2)) => (X4 = Y2)))) => ((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))))))).
% 3.76/1.96 thf(fact_29_inj__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),X2:A,Y:A]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((F2 @ X2) = (F2 @ Y)) = (X2 = Y)))))).
% 3.76/1.96 thf(fact_30_inj__def, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) = (! [X:A,Y3:A]: ((((F2 @ X) = (F2 @ Y3)) => (X = Y3)))))))).
% 3.76/1.96 thf(fact_31_inj__onD, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),X2:A,Y:A]: ((((((inj_on @ A) @ B) @ F2) @ A4) => (((F2 @ X2) = (F2 @ Y)) => ((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y) @ A4) => (X2 = Y)))))))).
% 3.76/1.96 thf(fact_32_inj__onI, axiom, (! [B:$tType,A:$tType,A4:(set @ A),F2:(A > B)]: (((! [X4:A,Y2:A]: (((((member @ A) @ X4) @ A4) => ((((member @ A) @ Y2) @ A4) => (((F2 @ X4) = (F2 @ Y2)) => (X4 = Y2)))))) => ((((inj_on @ A) @ B) @ F2) @ A4))))).
% 3.76/1.96 thf(fact_33_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F:(A > B),A5:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A5) => (! [Y3:A]: (((((member @ A) @ Y3) @ A5) => (((F @ X) = (F @ Y3)) => (X = Y3)))))))))))))).
% 3.76/1.96 thf(fact_34_inj__on__cong, axiom, (! [B:$tType,A:$tType,A4:(set @ A),F2:(A > B),G2:(A > B)]: (((! [A6:A]: (((((member @ A) @ A6) @ A4) => ((F2 @ A6) = (G2 @ A6))))) => (((((inj_on @ A) @ B) @ F2) @ A4) = ((((inj_on @ A) @ B) @ G2) @ A4)))))).
% 3.76/1.96 thf(fact_35_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),X2:A,Y:A]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y) @ A4) => (((F2 @ X2) = (F2 @ Y)) = (X2 = Y)))))))).
% 3.76/1.96 thf(fact_36_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),X2:A,Y:A]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((X2 != Y) => ((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y) @ A4) => ((F2 @ X2) != (F2 @ Y))))))))).
% 3.76/1.96 thf(fact_37_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A4:(set @ A),G2:(B > A),F2:(A > B)]: (((! [X4:A]: (((((member @ A) @ X4) @ A4) => ((G2 @ (F2 @ X4)) = X4)))) => ((((inj_on @ A) @ B) @ F2) @ A4))))).
% 3.76/1.96 thf(fact_38_inj__compose, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(A > B),G2:(C > A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((((inj_on @ C) @ A) @ G2) @ (top_top @ (set @ C))) => ((((inj_on @ C) @ B) @ (((((comp @ A) @ B) @ C) @ F2) @ G2)) @ (top_top @ (set @ C)))))))).
% 3.76/1.96 thf(fact_39_inj__on__imageI2, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(C > B),F2:(A > C),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F3) @ F2)) @ A4) => ((((inj_on @ A) @ C) @ F2) @ A4))))).
% 3.76/1.96 thf(fact_40_simple__path__linear__image__eq, axiom, (! [B:$tType,A:$tType]: ((((euclid925273238_space @ A) & (euclid925273238_space @ B)) => (! [F2:(A > B),G2:(real > A)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((path_simple_path @ B) @ (((((comp @ A) @ B) @ real) @ F2) @ G2)) = ((path_simple_path @ A) @ G2)))))))))).
% 3.76/1.96 thf(fact_41_fun_Oinj__map, axiom, (! [B:$tType,A:$tType,D:$tType,F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((inj_on @ (D > A)) @ (D > B)) @ ((((comp @ A) @ B) @ D) @ F2)) @ (top_top @ (set @ (D > A)))))))).
% 3.76/1.96 thf(fact_42_real__of__bounded__linear, axiom, (((real_V1632203528linear @ complex) @ a) @ (poinca179318102eal_of @ a))).
% 3.76/1.96 thf(fact_43_real__of__linear, axiom, (((real_Vector_linear @ complex) @ a) @ (poinca179318102eal_of @ a))).
% 3.76/1.96 thf(fact_44_iso__tuple__UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 3.76/1.96 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 3.76/1.96 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A4)))) = A4)))).
% 3.76/1.96 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P2 @ X4) = (Q @ X4)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 3.76/1.96 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G2:(A > B)]: (((! [X4:A]: (((F2 @ X4) = (G2 @ X4)))) => (F2 = G2))))).
% 3.76/1.96 thf(fact_49_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 3.76/1.96 thf(fact_50_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X:D]: ((top_top @ C)))))))).
% 3.76/1.96 thf(fact_51_bounded__linear_Ointro, axiom, (! [B:$tType,A:$tType]: ((((real_V55928688vector @ A) & (real_V55928688vector @ B)) => (! [F2:(A > B)]: (((((real_Vector_linear @ A) @ B) @ F2) => ((((real_V1072672427axioms @ A) @ B) @ F2) => (((real_V1632203528linear @ A) @ B) @ F2))))))))).
% 3.76/1.96 thf(fact_52_bounded__linear__def, axiom, (! [B:$tType,A:$tType]: ((((real_V55928688vector @ A) & (real_V55928688vector @ B)) => (((real_V1632203528linear @ A) @ B) = (^ [F:(A > B)]: (((((real_Vector_linear @ A) @ B) @ F) & (((real_V1072672427axioms @ A) @ B) @ F))))))))).
% 3.76/1.96 thf(fact_53_arc__linear__image__eq, axiom, (! [B:$tType,A:$tType]: ((((euclid925273238_space @ A) & (euclid925273238_space @ B)) => (! [F2:(A > B),G2:(real > A)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((path_arc @ B) @ (((((comp @ A) @ B) @ real) @ F2) @ G2)) = ((path_arc @ A) @ G2)))))))))).
% 3.76/1.96 thf(fact_54_real__of__inj, axiom, ((((inj_on @ complex) @ a) @ (poinca179318102eal_of @ a)) @ (top_top @ (set @ complex)))).
% 3.76/1.96 thf(fact_55_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 3.76/1.96 thf(fact_56_arc__imp__simple__path, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [G2:(real > A)]: ((((path_arc @ A) @ G2) => ((path_simple_path @ A) @ G2)))))))).
% 3.76/1.96 thf(fact_57_arc__distinct__ends, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [G2:(real > A)]: ((((path_arc @ A) @ G2) => (((path_pathfinish @ A) @ G2) != ((path_pathstart @ A) @ G2))))))))).
% 3.76/1.96 thf(fact_58_bounded__linear_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType]: ((((real_V55928688vector @ A) & (real_V55928688vector @ B)) => (! [F2:(A > B)]: (((((real_V1632203528linear @ A) @ B) @ F2) => (((real_V1072672427axioms @ A) @ B) @ F2)))))))).
% 3.76/1.96 thf(fact_59_UNIV__eq__I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X4:A]: ((((member @ A) @ X4) @ A4))) => ((top_top @ (set @ A)) = A4))))).
% 3.76/1.96 thf(fact_60_UNIV__witness, axiom, (! [A:$tType]: ((? [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A)))))))).
% 3.76/1.96 thf(fact_61_simple__path__imp__arc, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [G2:(real > A)]: ((((path_simple_path @ A) @ G2) => ((((path_pathfinish @ A) @ G2) != ((path_pathstart @ A) @ G2)) => ((path_arc @ A) @ G2))))))))).
% 3.76/1.96 thf(fact_62_simple__path__eq__arc, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [G2:(real > A)]: (((((path_pathfinish @ A) @ G2) != ((path_pathstart @ A) @ G2)) => (((path_simple_path @ A) @ G2) = ((path_arc @ A) @ G2))))))))).
% 3.76/1.96 thf(fact_63_simple__path__cases, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [G2:(real > A)]: ((((path_simple_path @ A) @ G2) => (((path_arc @ A) @ G2) | (((path_pathfinish @ A) @ G2) = ((path_pathstart @ A) @ G2)))))))))).
% 3.76/1.96 thf(fact_64_arc__simple__path, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => ((path_arc @ A) = (^ [G:(real > A)]: ((((path_simple_path @ A) @ G) & (((path_pathfinish @ A) @ G) != ((path_pathstart @ A) @ G)))))))))).
% 3.76/1.96 thf(fact_65_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G2:(B > C),F2:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G2) @ (((((comp @ A) @ B) @ D) @ F2) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G2) @ F2)) @ V))))).
% 3.76/1.96 thf(fact_66_complex__of__real__of, axiom, ((((((comp @ a) @ complex) @ complex) @ (poinca1531302022lex_of @ a)) @ (poinca179318102eal_of @ a)) = (id @ complex))).
% 3.76/1.96 thf(fact_67_real__of__complex__of, axiom, ((((((comp @ complex) @ a) @ a) @ (poinca179318102eal_of @ a)) @ (poinca1531302022lex_of @ a)) = (id @ a))).
% 3.76/1.96 thf(fact_68_real__of__bij, axiom, (((((bij_betw @ complex) @ a) @ (poinca179318102eal_of @ a)) @ (top_top @ (set @ complex))) @ (top_top @ (set @ a)))).
% 3.76/1.96 thf(fact_69_complex__of__bij, axiom, (((((bij_betw @ a) @ complex) @ (poinca1531302022lex_of @ a)) @ (top_top @ (set @ a))) @ (top_top @ (set @ complex)))).
% 3.76/1.96 thf(fact_70_linear__injective__left__inverse, axiom, (! [B:$tType,A:$tType]: ((((real_V1076094709vector @ A) & (real_V1076094709vector @ B)) => (! [F2:(A > B)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (? [G4:(B > A)]: (((((real_Vector_linear @ B) @ A) @ G4) & ((((((comp @ B) @ A) @ A) @ G4) @ F2) = (id @ A))))))))))))).
% 3.76/1.96 thf(fact_71_path__linear__image__eq, axiom, (! [B:$tType,A:$tType]: ((((euclid925273238_space @ A) & (euclid925273238_space @ B)) => (! [F2:(A > B),G2:(real > A)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((path_path @ B) @ (((((comp @ A) @ B) @ real) @ F2) @ G2)) = ((path_path @ A) @ G2)))))))))).
% 3.76/1.96 thf(fact_72_linear__inj__iff__eq__0, axiom, (! [B:$tType,A:$tType]: ((((real_V1076094709vector @ A) & (real_V1076094709vector @ B)) => (! [F2:(A > B)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) = (! [X:A]: ((((F2 @ X) = (zero_zero @ B)) => (X = (zero_zero @ A))))))))))))).
% 3.76/1.96 thf(fact_73_type__copy__map__cong0, axiom, (! [B:$tType,D:$tType,E:$tType,A:$tType,C:$tType,M:(B > A),G2:(C > B),X2:C,N:(D > A),H:(C > D),F2:(A > E)]: ((((M @ (G2 @ X2)) = (N @ (H @ X2))) => (((((((comp @ B) @ E) @ C) @ (((((comp @ A) @ E) @ B) @ F2) @ M)) @ G2) @ X2) = ((((((comp @ D) @ E) @ C) @ (((((comp @ A) @ E) @ D) @ F2) @ N)) @ H) @ X2)))))).
% 3.76/1.96 thf(fact_74_id__apply, axiom, (! [A:$tType]: (((id @ A) = (^ [X:A]: (X)))))).
% 3.76/1.96 thf(fact_75_top1I, axiom, (! [A:$tType,X2:A]: (((top_top @ (A > $o)) @ X2)))).
% 3.76/1.96 thf(fact_76_path__rectpath, axiom, (! [A2:complex,B2:complex]: (((path_path @ complex) @ ((path_rectpath @ A2) @ B2))))).
% 3.76/1.96 thf(fact_77_comp__id, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: (((((((comp @ A) @ B) @ A) @ F2) @ (id @ A)) = F2)))).
% 3.76/1.96 thf(fact_78_id__comp, axiom, (! [B:$tType,A:$tType,G2:(A > B)]: (((((((comp @ B) @ B) @ A) @ (id @ B)) @ G2) = G2)))).
% 3.76/1.96 thf(fact_79_fun_Omap__id, axiom, (! [A:$tType,D:$tType,T:(D > A)]: (((((((comp @ A) @ A) @ D) @ (id @ A)) @ T) = T)))).
% 3.76/1.96 thf(fact_80_bij__betw__id, axiom, (! [A:$tType,A4:(set @ A)]: ((((((bij_betw @ A) @ A) @ (id @ A)) @ A4) @ A4)))).
% 3.76/1.96 thf(fact_81_bij__id, axiom, (! [A:$tType]: ((((((bij_betw @ A) @ A) @ (id @ A)) @ (top_top @ (set @ A))) @ (top_top @ (set @ A)))))).
% 3.76/1.96 thf(fact_82_DEADID_Oin__rel, axiom, (! [B:$tType]: (((^ [Y4:B,Z:B]: ((Y4 = Z))) = (^ [A7:B,B3:B]: ((? [Z2:B]: (((((member @ B) @ Z2) @ (top_top @ (set @ B))) & ((((id @ B) @ Z2) = A7) & (((id @ B) @ Z2) = B3))))))))))).
% 3.76/1.96 thf(fact_83_top__conj_I1_J, axiom, (! [A:$tType,X2:A,P2:$o]: (((((top_top @ (A > $o)) @ X2) & P2) = P2)))).
% 3.76/1.96 thf(fact_84_top__conj_I2_J, axiom, (! [A:$tType,P2:$o,X2:A]: (((P2 & ((top_top @ (A > $o)) @ X2)) = P2)))).
% 3.76/1.96 thf(fact_85_fun_Omap__id0, axiom, (! [A:$tType,D:$tType]: ((((((comp @ A) @ A) @ D) @ (id @ A)) = (id @ (D > A)))))).
% 3.76/1.96 thf(fact_86_id__def, axiom, (! [A:$tType]: (((id @ A) = (^ [X:A]: (X)))))).
% 3.76/1.96 thf(fact_87_bij__betwE, axiom, (! [A:$tType,B:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => (! [X5:A]: (((((member @ A) @ X5) @ A4) => (((member @ B) @ (F2 @ X5)) @ B4)))))))).
% 3.76/1.96 thf(fact_88_eq__id__iff, axiom, (! [A:$tType,F2:(A > A)]: (((! [X:A]: (((F2 @ X) = X))) = (F2 = (id @ A)))))).
% 3.76/1.96 thf(fact_89_bij__betw__inv, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => (? [G4:(B > A)]: ((((((bij_betw @ B) @ A) @ G4) @ B4) @ A4))))))).
% 3.76/1.96 thf(fact_90_bij__betw__cong, axiom, (! [A:$tType,B:$tType,A4:(set @ A),F2:(A > B),G2:(A > B),A8:(set @ B)]: (((! [A6:A]: (((((member @ A) @ A6) @ A4) => ((F2 @ A6) = (G2 @ A6))))) => ((((((bij_betw @ A) @ B) @ F2) @ A4) @ A8) = (((((bij_betw @ A) @ B) @ G2) @ A4) @ A8)))))).
% 3.76/1.96 thf(fact_91_bij__betw__apply, axiom, (! [A:$tType,B:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B),A2:A]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => ((((member @ A) @ A2) @ A4) => (((member @ B) @ (F2 @ A2)) @ B4)))))).
% 3.76/1.96 thf(fact_92_bij__betw__id__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((((bij_betw @ A) @ A) @ (id @ A)) @ A4) @ B4) = (A4 = B4))))).
% 3.76/1.96 thf(fact_93_bij__betw__iff__bijections, axiom, (! [B:$tType,A:$tType]: ((((bij_betw @ A) @ B) = (^ [F:(A > B),A5:(set @ A),B5:(set @ B)]: ((? [G:(B > A)]: (((! [X:A]: (((((member @ A) @ X) @ A5) => ((((member @ B) @ (F @ X)) @ B5) & ((G @ (F @ X)) = X))))) & (! [X:B]: (((((member @ B) @ X) @ B5) => ((((member @ A) @ (G @ X)) @ A5) & ((F @ (G @ X)) = X)))))))))))))).
% 3.76/1.96 thf(fact_94_o__bij, axiom, (! [A:$tType,B:$tType,G2:(B > A),F2:(A > B)]: ((((((((comp @ B) @ A) @ A) @ G2) @ F2) = (id @ A)) => (((((((comp @ A) @ B) @ B) @ F2) @ G2) = (id @ B)) => (((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B)))))))).
% 3.76/1.96 thf(fact_95_bij__pointE, axiom, (! [B:$tType,A:$tType,F2:(A > B),Y:B]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (~ ((! [X4:A]: (((Y = (F2 @ X4)) => (~ ((! [X6:A]: (((Y = (F2 @ X6)) => (X6 = X4))))))))))))))).
% 3.76/1.96 thf(fact_96_bij__betw__trans, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B),G2:(B > C),C3:(set @ C)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => ((((((bij_betw @ B) @ C) @ G2) @ B4) @ C3) => (((((bij_betw @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G2) @ F2)) @ A4) @ C3)))))).
% 3.76/1.96 thf(fact_97_bij__betw__comp__iff, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(A > B),A4:(set @ A),A8:(set @ B),F3:(B > C),A9:(set @ C)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ A8) => ((((((bij_betw @ B) @ C) @ F3) @ A8) @ A9) = (((((bij_betw @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F3) @ F2)) @ A4) @ A9)))))).
% 3.76/1.96 thf(fact_98_bij__betw__imp__inj__on, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => ((((inj_on @ A) @ B) @ F2) @ A4))))).
% 3.76/1.96 thf(fact_99_pointfree__idE, axiom, (! [B:$tType,A:$tType,F2:(B > A),G2:(A > B),X2:A]: ((((((((comp @ B) @ A) @ A) @ F2) @ G2) = (id @ A)) => ((F2 @ (G2 @ X2)) = X2))))).
% 3.76/1.96 thf(fact_100_comp__eq__id__dest, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ B) @ B) @ A) @ (id @ B)) @ C2)) => ((A2 @ (B2 @ V)) = (C2 @ V)))))).
% 3.76/1.96 thf(fact_101_arc__imp__path, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [G2:(real > A)]: ((((path_arc @ A) @ G2) => ((path_path @ A) @ G2)))))))).
% 3.76/1.96 thf(fact_102_inj__on__id, axiom, (! [A:$tType,A4:(set @ A)]: (((((inj_on @ A) @ A) @ (id @ A)) @ A4)))).
% 3.76/1.96 thf(fact_103_simple__path__imp__path, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [G2:(real > A)]: ((((path_simple_path @ A) @ G2) => ((path_path @ A) @ G2)))))))).
% 3.76/1.96 thf(fact_104_module__hom__id, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (((real_Vector_linear @ A) @ A) @ (id @ A)))))).
% 3.76/1.96 thf(fact_105_linear__simps_I3_J, axiom, (! [A:$tType,B:$tType]: ((((real_V55928688vector @ B) & (real_V55928688vector @ A)) => (! [F2:(A > B)]: (((((real_V1632203528linear @ A) @ B) @ F2) => ((F2 @ (zero_zero @ A)) = (zero_zero @ B))))))))).
% 3.76/1.96 thf(fact_106_linear__0, axiom, (! [A:$tType,B:$tType]: ((((real_V1076094709vector @ B) & (real_V1076094709vector @ A)) => (! [F2:(A > B)]: (((((real_Vector_linear @ A) @ B) @ F2) => ((F2 @ (zero_zero @ A)) = (zero_zero @ B))))))))).
% 3.76/1.96 thf(fact_107_pathstart__def, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => ((path_pathstart @ A) = (^ [G:(real > A)]: ((G @ (zero_zero @ real))))))))).
% 3.76/1.96 thf(fact_108_bij__comp, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(A > B),G2:(B > C)]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((((bij_betw @ B) @ C) @ G2) @ (top_top @ (set @ B))) @ (top_top @ (set @ C))) => (((((bij_betw @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G2) @ F2)) @ (top_top @ (set @ A))) @ (top_top @ (set @ C)))))))).
% 3.76/1.96 thf(fact_109_bij__is__inj, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))))))).
% 3.76/1.96 thf(fact_110_right__inverse__linear, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [F2:(A > A),G2:(A > A)]: (((((real_Vector_linear @ A) @ A) @ F2) => (((((((comp @ A) @ A) @ A) @ F2) @ G2) = (id @ A)) => (((real_Vector_linear @ A) @ A) @ G2))))))))).
% 3.76/1.96 thf(fact_111_linear__inverse__left, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [F2:(A > A),F3:(A > A)]: (((((real_Vector_linear @ A) @ A) @ F2) => ((((real_Vector_linear @ A) @ A) @ F3) => (((((((comp @ A) @ A) @ A) @ F2) @ F3) = (id @ A)) = ((((((comp @ A) @ A) @ A) @ F3) @ F2) = (id @ A))))))))))).
% 3.76/1.96 thf(fact_112_left__inverse__linear, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [F2:(A > A),G2:(A > A)]: (((((real_Vector_linear @ A) @ A) @ F2) => (((((((comp @ A) @ A) @ A) @ G2) @ F2) = (id @ A)) => (((real_Vector_linear @ A) @ A) @ G2))))))))).
% 3.76/1.96 thf(fact_113_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A))))))))).
% 3.76/1.96 thf(fact_114_left__right__inverse__eq, axiom, (! [A:$tType,B:$tType,F2:(B > A),G2:(A > B),H:(B > A)]: ((((((((comp @ B) @ A) @ A) @ F2) @ G2) = (id @ A)) => (((((((comp @ A) @ B) @ B) @ G2) @ H) = (id @ B)) => (F2 = H)))))).
% 3.76/1.96 thf(fact_115_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 3.76/1.96 thf(fact_116_isomorphism__expand, axiom, (! [A:$tType,B:$tType,F2:(B > A),G2:(A > B)]: (((((((((comp @ B) @ A) @ A) @ F2) @ G2) = (id @ A)) & ((((((comp @ A) @ B) @ B) @ G2) @ F2) = (id @ B))) = ((! [X:A]: (((F2 @ (G2 @ X)) = X))) & (! [X:B]: (((G2 @ (F2 @ X)) = X)))))))).
% 3.76/1.96 thf(fact_117_bijI_H, axiom, (! [A:$tType,B:$tType,F2:(A > B)]: (((! [X4:A,Y2:A]: ((((F2 @ X4) = (F2 @ Y2)) = (X4 = Y2)))) => ((! [Y2:B]: ((? [X5:A]: ((Y2 = (F2 @ X5)))))) => (((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B)))))))).
% 3.76/1.96 thf(fact_118_bij__iff, axiom, (! [A:$tType,B:$tType,F2:(A > B)]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (! [X:B]: ((? [Y3:A]: ((((F2 @ Y3) = X) & (! [Z2:A]: ((((F2 @ Z2) = X) => (Z2 = Y3))))))))))))).
% 3.76/1.96 thf(fact_119_c1__on__open__R2_Ocomplex__of__real__of, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => ((((((comp @ A) @ complex) @ complex) @ (poinca1531302022lex_of @ A)) @ (poinca179318102eal_of @ A)) = (id @ complex))))))))).
% 3.76/1.96 thf(fact_120_c1__on__open__R2_Oreal__of__bij, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => (((((bij_betw @ complex) @ A) @ (poinca179318102eal_of @ A)) @ (top_top @ (set @ complex))) @ (top_top @ (set @ A)))))))))).
% 3.76/1.96 thf(fact_121_c1__on__open__R2_Oreal__of__bounded__linear, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => (((real_V1632203528linear @ complex) @ A) @ (poinca179318102eal_of @ A))))))))).
% 3.76/1.96 thf(fact_122_c1__on__open__R2_Ocomplex__of__bounded__linear, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => (((real_V1632203528linear @ A) @ complex) @ (poinca1531302022lex_of @ A))))))))).
% 3.76/1.96 thf(fact_123_c1__on__open__R2_Ocomplex__of__linear, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => (((real_Vector_linear @ A) @ complex) @ (poinca1531302022lex_of @ A))))))))).
% 3.76/1.96 thf(fact_124_c1__on__open__R2_Oreal__of__linear, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => (((real_Vector_linear @ complex) @ A) @ (poinca179318102eal_of @ A))))))))).
% 3.76/1.96 thf(fact_125_comp__apply__eq, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,F2:(B > A),G2:(C > B),X2:C,H:(D > A),K:(C > D)]: ((((F2 @ (G2 @ X2)) = (H @ (K @ X2))) => (((((((comp @ B) @ A) @ C) @ F2) @ G2) @ X2) = ((((((comp @ D) @ A) @ C) @ H) @ K) @ X2)))))).
% 3.76/1.96 thf(fact_126_c1__on__open__R2_Oreal__of__inj, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => ((((inj_on @ complex) @ A) @ (poinca179318102eal_of @ A)) @ (top_top @ (set @ complex)))))))))).
% 3.76/1.96 thf(fact_127_c1__on__open__R2_Oreal__of__complex__of, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => ((((((comp @ complex) @ A) @ A) @ (poinca179318102eal_of @ A)) @ (poinca1531302022lex_of @ A)) = (id @ A))))))))).
% 3.76/1.96 thf(fact_128_c1__on__open__R2_Ocomplex__of__bij, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (! [F2:(A > A),F3:(A > ((bounde2145540817linfun @ A) @ A)),X7:(set @ A)]: ((((((poinca1076805489pen_R2 @ A) @ F2) @ F3) @ X7) => (((((bij_betw @ A) @ complex) @ (poinca1531302022lex_of @ A)) @ (top_top @ (set @ A))) @ (top_top @ (set @ complex)))))))))).
% 3.76/1.96 thf(fact_129_the__inv__f__o__f__id, axiom, (! [B:$tType,A:$tType,F2:(A > B),Z3:A]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((((((comp @ B) @ A) @ A) @ ((((the_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2)) @ F2) @ Z3) = ((id @ A) @ Z3)))))).
% 3.76/1.96 thf(fact_130_linear__surjective__right__inverse, axiom, (! [A:$tType,B:$tType]: ((((real_V1076094709vector @ B) & (real_V1076094709vector @ A)) => (! [F2:(A > B)]: (((((real_Vector_linear @ A) @ B) @ F2) => ((((((image @ A) @ B) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => (? [G4:(B > A)]: (((((real_Vector_linear @ B) @ A) @ G4) & ((((((comp @ A) @ B) @ B) @ F2) @ G4) = (id @ B))))))))))))).
% 3.76/1.96 thf(fact_131_iso__tuple__update__accessor__eq__assist__idI, axiom, (! [A:$tType,V3:A,F2:(A > A),V:A]: (((V3 = (F2 @ V)) => ((((((((iso_tu2011167877assist @ A) @ A) @ (id @ (A > A))) @ (id @ A)) @ V) @ F2) @ V3) @ V))))).
% 3.76/1.96 thf(fact_132_bij__swap__compose__bij, axiom, (! [A:$tType,B:$tType,P:(A > B),A2:B,B2:B]: (((((((bij_betw @ A) @ B) @ P) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((bij_betw @ A) @ B) @ (((((comp @ B) @ B) @ A) @ (((((swap @ B) @ B) @ A2) @ B2) @ (id @ B))) @ P)) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))))))).
% 3.76/1.96 thf(fact_133_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F2:(B > A),X2:B,A4:(set @ B)]: (((B2 = (F2 @ X2)) => ((((member @ B) @ X2) @ A4) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F2) @ A4))))))).
% 3.76/1.96 thf(fact_134_swap__nilpotent, axiom, (! [B:$tType,A:$tType,A2:A,B2:A,F2:(A > B)]: (((((((swap @ A) @ B) @ A2) @ B2) @ (((((swap @ A) @ B) @ A2) @ B2) @ F2)) = F2)))).
% 3.76/1.96 thf(fact_135_swap__self, axiom, (! [B:$tType,A:$tType,A2:A,F2:(A > B)]: (((((((swap @ A) @ B) @ A2) @ A2) @ F2) = F2)))).
% 3.76/1.96 thf(fact_136_swap__apply_I3_J, axiom, (! [A:$tType,B:$tType,C2:B,A2:B,B2:B,F2:(B > A)]: (((C2 != A2) => ((C2 != B2) => (((((((swap @ B) @ A) @ A2) @ B2) @ F2) @ C2) = (F2 @ C2))))))).
% 3.76/1.96 thf(fact_137_image__id, axiom, (! [A:$tType]: (((((image @ A) @ A) @ (id @ A)) = (id @ (set @ A)))))).
% 3.76/1.96 thf(fact_138_swap__image__eq, axiom, (! [B:$tType,A:$tType,A2:A,A4:(set @ A),B2:A,F2:(A > B)]: (((((member @ A) @ A2) @ A4) => ((((member @ A) @ B2) @ A4) => (((((image @ A) @ B) @ (((((swap @ A) @ B) @ A2) @ B2) @ F2)) @ A4) = ((((image @ A) @ B) @ F2) @ A4))))))).
% 3.76/1.96 thf(fact_139_inj__on__swap__iff, axiom, (! [B:$tType,A:$tType,A2:A,A4:(set @ A),B2:A,F2:(A > B)]: (((((member @ A) @ A2) @ A4) => ((((member @ A) @ B2) @ A4) => (((((inj_on @ A) @ B) @ (((((swap @ A) @ B) @ A2) @ B2) @ F2)) @ A4) = ((((inj_on @ A) @ B) @ F2) @ A4))))))).
% 3.76/1.96 thf(fact_140_bij__betw__swap__iff, axiom, (! [A:$tType,B:$tType,X2:A,A4:(set @ A),Y:A,F2:(A > B),B4:(set @ B)]: (((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y) @ A4) => ((((((bij_betw @ A) @ B) @ (((((swap @ A) @ B) @ X2) @ Y) @ F2)) @ A4) @ B4) = (((((bij_betw @ A) @ B) @ F2) @ A4) @ B4))))))).
% 3.76/1.96 thf(fact_141_swap__comp__involutory, axiom, (! [B:$tType,A:$tType,A2:A,B2:A]: (((((((comp @ (A > B)) @ (A > B)) @ (A > B)) @ ((((swap @ A) @ B) @ A2) @ B2)) @ ((((swap @ A) @ B) @ A2) @ B2)) = (id @ (A > B)))))).
% 3.76/1.96 thf(fact_142_surj__swap__iff, axiom, (! [B:$tType,A:$tType,A2:B,B2:B,F2:(B > A)]: (((((((image @ B) @ A) @ (((((swap @ B) @ A) @ A2) @ B2) @ F2)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.76/1.96 thf(fact_143_bij__swap__iff, axiom, (! [A:$tType,B:$tType,A2:A,B2:A,F2:(A > B)]: (((((((bij_betw @ A) @ B) @ (((((swap @ A) @ B) @ A2) @ B2) @ F2)) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))))))).
% 3.76/1.96 thf(fact_144_swap__id__idempotent, axiom, (! [A:$tType,A2:A,B2:A]: (((((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))) = (id @ A))))).
% 3.76/1.96 thf(fact_145_the__inv__into__onto, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A4) => (((((image @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A4) @ F2)) @ ((((image @ A) @ B) @ F2) @ A4)) = A4))))).
% 3.76/1.96 thf(fact_146_swap__id__eq, axiom, (! [A:$tType,X2:A,A2:A,B2:A]: ((((X2 = A2) => (((((((swap @ A) @ A) @ A2) @ B2) @ (id @ A)) @ X2) = B2)) & ((X2 != A2) => (((X2 = B2) => (((((((swap @ A) @ A) @ A2) @ B2) @ (id @ A)) @ X2) = A2)) & ((X2 != B2) => (((((((swap @ A) @ A) @ A2) @ B2) @ (id @ A)) @ X2) = X2)))))))).
% 3.76/1.96 thf(fact_147_comp__swap, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(C > B),A2:A,B2:A,G2:(A > C)]: (((((((comp @ C) @ B) @ A) @ F2) @ (((((swap @ A) @ C) @ A2) @ B2) @ G2)) = (((((swap @ A) @ B) @ A2) @ B2) @ (((((comp @ C) @ B) @ A) @ F2) @ G2)))))).
% 3.76/1.96 thf(fact_148_the__inv__into__comp, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(A > B),G2:(C > A),A4:(set @ C),X2:B]: ((((((inj_on @ A) @ B) @ F2) @ ((((image @ C) @ A) @ G2) @ A4)) => (((((inj_on @ C) @ A) @ G2) @ A4) => ((((member @ B) @ X2) @ ((((image @ A) @ B) @ F2) @ ((((image @ C) @ A) @ G2) @ A4))) => ((((((the_inv_into @ C) @ B) @ A4) @ (((((comp @ A) @ B) @ C) @ F2) @ G2)) @ X2) = ((((((comp @ A) @ C) @ B) @ ((((the_inv_into @ C) @ A) @ A4) @ G2)) @ ((((the_inv_into @ A) @ B) @ ((((image @ C) @ A) @ G2) @ A4)) @ F2)) @ X2)))))))).
% 3.76/1.96 thf(fact_149_image__eq__imp__comp, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F2:(B > A),A4:(set @ B),G2:(C > A),B4:(set @ C),H:(A > D)]: (((((((image @ B) @ A) @ F2) @ A4) = ((((image @ C) @ A) @ G2) @ B4)) => (((((image @ B) @ D) @ (((((comp @ A) @ D) @ B) @ H) @ F2)) @ A4) = ((((image @ C) @ D) @ (((((comp @ A) @ D) @ C) @ H) @ G2)) @ B4)))))).
% 3.76/1.96 thf(fact_150_image__comp, axiom, (! [B:$tType,A:$tType,C:$tType,F2:(B > A),G2:(C > B),R:(set @ C)]: ((((((image @ B) @ A) @ F2) @ ((((image @ C) @ B) @ G2) @ R)) = ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F2) @ G2)) @ R))))).
% 3.76/1.96 thf(fact_151_bij__betw__imp__surj__on, axiom, (! [A:$tType,B:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => (((((image @ A) @ B) @ F2) @ A4) = B4))))).
% 3.76/1.96 thf(fact_152_surj__imp__surj__swap, axiom, (! [B:$tType,A:$tType,F2:(B > A),A2:B,B2:B]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ (((((swap @ B) @ A) @ A2) @ B2) @ F2)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.76/1.96 thf(fact_153_inj__on__the__inv__into, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((((inj_on @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A4) @ F2)) @ ((((image @ A) @ B) @ F2) @ A4)))))).
% 3.76/1.96 thf(fact_154_f__the__inv__into__f, axiom, (! [A:$tType,B:$tType,F2:(A > B),A4:(set @ A),Y:B]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((((member @ B) @ Y) @ ((((image @ A) @ B) @ F2) @ A4)) => ((F2 @ (((((the_inv_into @ A) @ B) @ A4) @ F2) @ Y)) = Y)))))).
% 3.76/1.96 thf(fact_155_swap__commute, axiom, (! [B:$tType,A:$tType]: ((((swap @ A) @ B) = (^ [A7:A,B3:A]: (((((swap @ A) @ B) @ B3) @ A7))))))).
% 3.76/1.96 thf(fact_156_swap__triple, axiom, (! [B:$tType,A:$tType,A2:A,C2:A,B2:A,F2:(A > B)]: (((A2 != C2) => ((B2 != C2) => ((((((swap @ A) @ B) @ A2) @ B2) @ (((((swap @ A) @ B) @ B2) @ C2) @ (((((swap @ A) @ B) @ A2) @ B2) @ F2))) = (((((swap @ A) @ B) @ A2) @ C2) @ F2))))))).
% 3.76/1.96 thf(fact_157_iso__tuple__update__accessor__eq__assist__triv, axiom, (! [B:$tType,A:$tType,Upd:((A > A) > (B > B)),Ac:(B > A),V:B,F2:(A > A),V3:B,X2:A]: ((((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac) @ V) @ F2) @ V3) @ X2) => ((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac) @ V) @ F2) @ V3) @ X2))))).
% 3.76/1.96 thf(fact_158_update__accessor__accessor__eqE, axiom, (! [B:$tType,A:$tType,Upd:((A > A) > (B > B)),Ac:(B > A),V:B,F2:(A > A),V3:B,X2:A]: ((((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac) @ V) @ F2) @ V3) @ X2) => ((Ac @ V) = X2))))).
% 3.76/1.96 thf(fact_159_update__accessor__updator__eqE, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac:(B > A),V:B,F2:(A > A),V3:B,X2:A]: ((((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac) @ V) @ F2) @ V3) @ X2) => (((Upd @ F2) @ V) = V3))))).
% 3.76/1.96 thf(fact_160_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X2:A,A4:(set @ A),B2:B,F2:(A > B)]: (((((member @ A) @ X2) @ A4) => ((B2 = (F2 @ X2)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F2) @ A4))))))).
% 3.76/1.96 thf(fact_161_ball__imageD, axiom, (! [A:$tType,B:$tType,F2:(B > A),A4:(set @ B),P2:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F2) @ A4)) => (P2 @ X4)))) => (! [X5:B]: (((((member @ B) @ X5) @ A4) => (P2 @ (F2 @ X5))))))))).
% 3.76/1.96 thf(fact_162_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F2:(A > B),G2:(A > B)]: (((M = N) => ((! [X4:A]: (((((member @ A) @ X4) @ N) => ((F2 @ X4) = (G2 @ X4))))) => (((((image @ A) @ B) @ F2) @ M) = ((((image @ A) @ B) @ G2) @ N))))))).
% 3.76/1.96 thf(fact_163_bex__imageD, axiom, (! [A:$tType,B:$tType,F2:(B > A),A4:(set @ B),P2:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F2) @ A4)) & (P2 @ X5)))) => (? [X4:B]: (((((member @ B) @ X4) @ A4) & (P2 @ (F2 @ X4))))))))).
% 3.76/1.96 thf(fact_164_image__iff, axiom, (! [A:$tType,B:$tType,Z3:A,F2:(B > A),A4:(set @ B)]: (((((member @ A) @ Z3) @ ((((image @ B) @ A) @ F2) @ A4)) = (? [X:B]: (((((member @ B) @ X) @ A4) & (Z3 = (F2 @ X))))))))).
% 3.76/1.96 thf(fact_165_imageI, axiom, (! [B:$tType,A:$tType,X2:A,A4:(set @ A),F2:(A > B)]: (((((member @ A) @ X2) @ A4) => (((member @ B) @ (F2 @ X2)) @ ((((image @ A) @ B) @ F2) @ A4)))))).
% 3.76/1.96 thf(fact_166_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A4:(set @ A),G2:(A > B),F2:(A > A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A4) => (! [Xa:A]: (((((member @ A) @ Xa) @ A4) => (((G2 @ (F2 @ X4)) = (G2 @ (F2 @ Xa))) = ((G2 @ X4) = (G2 @ Xa))))))))) => (((((inj_on @ A) @ A) @ F2) @ A4) => (((((inj_on @ A) @ B) @ G2) @ ((((image @ A) @ A) @ F2) @ A4)) = ((((inj_on @ A) @ B) @ G2) @ A4))))))).
% 3.76/1.96 thf(fact_167_surjD, axiom, (! [A:$tType,B:$tType,F2:(B > A),Y:A]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X4:B]: ((Y = (F2 @ X4)))))))).
% 3.76/1.96 thf(fact_168_surjE, axiom, (! [A:$tType,B:$tType,F2:(B > A),Y:A]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X4:B]: ((Y != (F2 @ X4)))))))))).
% 3.76/1.96 thf(fact_169_surjI, axiom, (! [B:$tType,A:$tType,G2:(B > A),F2:(A > B)]: (((! [X4:A]: (((G2 @ (F2 @ X4)) = X4))) => (((((image @ B) @ A) @ G2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.76/1.96 thf(fact_170_surj__def, axiom, (! [B:$tType,A:$tType,F2:(B > A)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y3:A]: ((? [X:B]: ((Y3 = (F2 @ X)))))))))).
% 3.76/1.96 thf(fact_171_rangeI, axiom, (! [A:$tType,B:$tType,F2:(B > A),X2:B]: ((((member @ A) @ (F2 @ X2)) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))))))).
% 3.76/1.96 thf(fact_172_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F2:(B > A),X2:B]: (((B2 = (F2 @ X2)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B)))))))).
% 3.76/1.96 thf(fact_173_inj__on__imp__inj__on__swap, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),A2:A,B2:A]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((((member @ A) @ A2) @ A4) => ((((member @ A) @ B2) @ A4) => ((((inj_on @ A) @ B) @ (((((swap @ A) @ B) @ A2) @ B2) @ F2)) @ A4))))))).
% 3.76/1.96 thf(fact_174_the__inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),X2:A,Y:B]: ((((((inj_on @ A) @ B) @ F2) @ A4) => (((F2 @ X2) = Y) => ((((member @ A) @ X2) @ A4) => ((((((the_inv_into @ A) @ B) @ A4) @ F2) @ Y) = X2))))))).
% 3.76/1.96 thf(fact_175_the__inv__into__f__f, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),X2:A]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((((member @ A) @ X2) @ A4) => ((((((the_inv_into @ A) @ B) @ A4) @ F2) @ (F2 @ X2)) = X2)))))).
% 3.76/1.96 thf(fact_176_f__the__inv__into__f__bij__betw, axiom, (! [A:$tType,B:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B),X2:B]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => (((member @ B) @ X2) @ B4)) => ((F2 @ (((((the_inv_into @ A) @ B) @ A4) @ F2) @ X2)) = X2)))))).
% 3.76/1.96 thf(fact_177_bij__betw__the__inv__into, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => (((((bij_betw @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A4) @ F2)) @ B4) @ A4))))).
% 3.76/1.96 thf(fact_178_swap__general, axiom, (! [A:$tType,A2:A,B2:A,C2:A,D2:A]: (((A2 != B2) => ((C2 != D2) => (((((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))) @ (((((swap @ A) @ A) @ C2) @ D2) @ (id @ A))) = (id @ A)) | (? [X4:A,Y2:A,Z4:A]: (((X4 != A2) & ((Y2 != A2) & ((Z4 != A2) & ((X4 != Y2) & ((((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))) @ (((((swap @ A) @ A) @ C2) @ D2) @ (id @ A))) = (((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ X4) @ Y2) @ (id @ A))) @ (((((swap @ A) @ A) @ A2) @ Z4) @ (id @ A)))))))))))))))).
% 3.76/1.96 thf(fact_179_swap__id__common, axiom, (! [A:$tType,A2:A,C2:A,B2:A]: (((A2 != C2) => ((B2 != C2) => ((((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))) @ (((((swap @ A) @ A) @ A2) @ C2) @ (id @ A))) = (((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ B2) @ C2) @ (id @ A))) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))))))))).
% 3.76/1.96 thf(fact_180_swap__id__common_H, axiom, (! [A:$tType,A2:A,B2:A,C2:A]: (((A2 != B2) => ((A2 != C2) => ((((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ A2) @ C2) @ (id @ A))) @ (((((swap @ A) @ A) @ B2) @ C2) @ (id @ A))) = (((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ B2) @ C2) @ (id @ A))) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))))))))).
% 3.76/1.96 thf(fact_181_swap__id__independent, axiom, (! [A:$tType,A2:A,C2:A,D2:A,B2:A]: (((A2 != C2) => ((A2 != D2) => ((B2 != C2) => ((B2 != D2) => ((((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))) @ (((((swap @ A) @ A) @ C2) @ D2) @ (id @ A))) = (((((comp @ A) @ A) @ A) @ (((((swap @ A) @ A) @ C2) @ D2) @ (id @ A))) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))))))))))).
% 3.76/1.96 thf(fact_182_fun_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,D:$tType,X2:(D > A),Xa2:(D > A),F2:(A > B),Fa:(A > B)]: (((! [Z4:A,Za:A]: (((((member @ A) @ Z4) @ ((((image @ D) @ A) @ X2) @ (top_top @ (set @ D)))) => ((((member @ A) @ Za) @ ((((image @ D) @ A) @ Xa2) @ (top_top @ (set @ D)))) => (((F2 @ Z4) = (Fa @ Za)) => (Z4 = Za)))))) => (((((((comp @ A) @ B) @ D) @ F2) @ X2) = (((((comp @ A) @ B) @ D) @ Fa) @ Xa2)) => (X2 = Xa2)))))).
% 3.76/1.96 thf(fact_183_fun_Omap__cong0, axiom, (! [B:$tType,A:$tType,D:$tType,X2:(D > A),F2:(A > B),G2:(A > B)]: (((! [Z4:A]: (((((member @ A) @ Z4) @ ((((image @ D) @ A) @ X2) @ (top_top @ (set @ D)))) => ((F2 @ Z4) = (G2 @ Z4))))) => ((((((comp @ A) @ B) @ D) @ F2) @ X2) = (((((comp @ A) @ B) @ D) @ G2) @ X2)))))).
% 3.76/1.96 thf(fact_184_fun_Omap__cong, axiom, (! [B:$tType,A:$tType,D:$tType,X2:(D > A),Ya:(D > A),F2:(A > B),G2:(A > B)]: (((X2 = Ya) => ((! [Z4:A]: (((((member @ A) @ Z4) @ ((((image @ D) @ A) @ Ya) @ (top_top @ (set @ D)))) => ((F2 @ Z4) = (G2 @ Z4))))) => ((((((comp @ A) @ B) @ D) @ F2) @ X2) = (((((comp @ A) @ B) @ D) @ G2) @ Ya))))))).
% 3.76/1.96 thf(fact_185_fun_Oset__map, axiom, (! [B:$tType,A:$tType,D:$tType,F2:(A > B),V:(D > A)]: ((((((image @ D) @ B) @ (((((comp @ A) @ B) @ D) @ F2) @ V)) @ (top_top @ (set @ D))) = ((((image @ A) @ B) @ F2) @ ((((image @ D) @ A) @ V) @ (top_top @ (set @ D)))))))).
% 3.76/1.96 thf(fact_186_comp__surj, axiom, (! [B:$tType,A:$tType,C:$tType,F2:(B > A),G2:(A > C)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((((image @ A) @ C) @ G2) @ (top_top @ (set @ A))) = (top_top @ (set @ C))) => (((((image @ B) @ C) @ (((((comp @ A) @ C) @ B) @ G2) @ F2)) @ (top_top @ (set @ B))) = (top_top @ (set @ C)))))))).
% 3.76/1.96 thf(fact_187_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A2:A,A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((member @ B) @ (F2 @ A2)) @ ((((image @ A) @ B) @ F2) @ A4)) = (((member @ A) @ A2) @ A4)))))).
% 3.76/1.96 thf(fact_188_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),B4:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F2) @ A4) = ((((image @ A) @ B) @ F2) @ B4)) = (A4 = B4)))))).
% 3.76/1.96 thf(fact_189_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),B2:B]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((member @ B) @ B2) @ ((((image @ A) @ B) @ F2) @ (top_top @ (set @ A)))) = (? [X:A]: (((B2 = (F2 @ X)) & (! [Y3:A]: (((B2 = (F2 @ Y3)) => (Y3 = X)))))))))))).
% 3.76/1.96 thf(fact_190_surj__id, axiom, (! [A:$tType]: ((((((image @ A) @ A) @ (id @ A)) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 3.76/1.96 thf(fact_191_comp__inj__on__iff, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(A > B),A4:(set @ A),F3:(B > C)]: ((((((inj_on @ A) @ B) @ F2) @ A4) => (((((inj_on @ B) @ C) @ F3) @ ((((image @ A) @ B) @ F2) @ A4)) = ((((inj_on @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F3) @ F2)) @ A4)))))).
% 3.76/1.96 thf(fact_192_inj__on__imageI, axiom, (! [B:$tType,C:$tType,A:$tType,G2:(C > B),F2:(A > C),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ (((((comp @ C) @ B) @ A) @ G2) @ F2)) @ A4) => ((((inj_on @ C) @ B) @ G2) @ ((((image @ A) @ C) @ F2) @ A4)))))).
% 3.76/1.96 thf(fact_193_comp__inj__on, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(A > B),A4:(set @ A),G2:(B > C)]: ((((((inj_on @ A) @ B) @ F2) @ A4) => (((((inj_on @ B) @ C) @ G2) @ ((((image @ A) @ B) @ F2) @ A4)) => ((((inj_on @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G2) @ F2)) @ A4)))))).
% 3.76/1.96 thf(fact_194_bij__betw__imp__surj, axiom, (! [A:$tType,B:$tType,F2:(A > B),A4:(set @ A)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ (top_top @ (set @ B))) => (((((image @ A) @ B) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ B))))))).
% 3.76/1.96 thf(fact_195_bij__is__surj, axiom, (! [A:$tType,B:$tType,F2:(A > B)]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((image @ A) @ B) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ B))))))).
% 3.76/1.96 thf(fact_196_inj__on__imp__bij__betw, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A4) => (((((bij_betw @ A) @ B) @ F2) @ A4) @ ((((image @ A) @ B) @ F2) @ A4)))))).
% 3.76/1.96 thf(fact_197_bij__betw__imageI, axiom, (! [A:$tType,B:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((((((image @ A) @ B) @ F2) @ A4) = B4) => (((((bij_betw @ A) @ B) @ F2) @ A4) @ B4)))))).
% 3.76/1.96 thf(fact_198_bij__betw__def, axiom, (! [B:$tType,A:$tType]: ((((bij_betw @ A) @ B) = (^ [F:(A > B),A5:(set @ A),B5:(set @ B)]: ((((((inj_on @ A) @ B) @ F) @ A5) & (((((image @ A) @ B) @ F) @ A5) = B5)))))))).
% 3.76/1.96 thf(fact_199_the__inv__f__f, axiom, (! [B:$tType,A:$tType,F2:(A > B),X2:A]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((((the_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2) @ (F2 @ X2)) = X2))))).
% 3.76/1.96 thf(fact_200_bij__def, axiom, (! [A:$tType,B:$tType,F2:(A > B)]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) & (((((image @ A) @ B) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ B)))))))).
% 3.76/1.96 thf(fact_201_bijI, axiom, (! [A:$tType,B:$tType,F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => (((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B)))))))).
% 3.76/1.96 thf(fact_202_linear__surj__imp__inj, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [F2:(A > A)]: (((((real_Vector_linear @ A) @ A) @ F2) => ((((((image @ A) @ A) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ A))) => ((((inj_on @ A) @ A) @ F2) @ (top_top @ (set @ A))))))))))).
% 3.76/1.96 thf(fact_203_linear__inj__imp__surj, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [F2:(A > A)]: (((((real_Vector_linear @ A) @ A) @ F2) => (((((inj_on @ A) @ A) @ F2) @ (top_top @ (set @ A))) => (((((image @ A) @ A) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))))).
% 3.76/1.96 thf(fact_204_surj__fun__eq, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(B > A),X7:(set @ B),G1:(A > C),G22:(A > C)]: (((((((image @ B) @ A) @ F2) @ X7) = (top_top @ (set @ A))) => ((! [X4:B]: (((((member @ B) @ X4) @ X7) => (((((((comp @ A) @ C) @ B) @ G1) @ F2) @ X4) = ((((((comp @ A) @ C) @ B) @ G22) @ F2) @ X4))))) => (G1 = G22)))))).
% 3.76/1.96 thf(fact_205_Inf_OINF__id__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A4:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (id @ A)) @ A4)) = (Inf @ A4))))).
% 3.76/1.96 thf(fact_206_Sup_OSUP__id__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A4:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (id @ A)) @ A4)) = (Sup @ A4))))).
% 3.76/1.96 thf(fact_207_Inf_OINF__image, axiom, (! [B:$tType,A:$tType,C:$tType,Inf:((set @ A) > A),G2:(B > A),F2:(C > B),A4:(set @ C)]: (((Inf @ ((((image @ B) @ A) @ G2) @ ((((image @ C) @ B) @ F2) @ A4))) = (Inf @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G2) @ F2)) @ A4)))))).
% 3.76/1.96 thf(fact_208_bij__betwI_H, axiom, (! [A:$tType,B:$tType,X7:(set @ A),F2:(A > B),Y5:(set @ B)]: (((! [X4:A]: (((((member @ A) @ X4) @ X7) => (! [Y2:A]: (((((member @ A) @ Y2) @ X7) => (((F2 @ X4) = (F2 @ Y2)) = (X4 = Y2)))))))) => ((! [X4:A]: (((((member @ A) @ X4) @ X7) => (((member @ B) @ (F2 @ X4)) @ Y5)))) => ((! [Y2:B]: (((((member @ B) @ Y2) @ Y5) => (? [X5:A]: (((((member @ A) @ X5) @ X7) & (Y2 = (F2 @ X5)))))))) => (((((bij_betw @ A) @ B) @ F2) @ X7) @ Y5))))))).
% 3.76/1.96 thf(fact_209_Sup_OSUP__image, axiom, (! [B:$tType,A:$tType,C:$tType,Sup:((set @ A) > A),G2:(B > A),F2:(C > B),A4:(set @ C)]: (((Sup @ ((((image @ B) @ A) @ G2) @ ((((image @ C) @ B) @ F2) @ A4))) = (Sup @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G2) @ F2)) @ A4)))))).
% 3.76/1.96 thf(fact_210_inj__adjoint__iff__surj, axiom, (! [M2:$tType,N2:$tType]: ((((euclid925273238_space @ N2) & (euclid925273238_space @ M2)) => (! [F2:(M2 > N2)]: (((((real_Vector_linear @ M2) @ N2) @ F2) => (((((inj_on @ N2) @ M2) @ (((linear_adjoint @ M2) @ N2) @ F2)) @ (top_top @ (set @ N2))) = (((((image @ M2) @ N2) @ F2) @ (top_top @ (set @ M2))) = (top_top @ (set @ N2))))))))))).
% 3.76/1.96 thf(fact_211_surj__adjoint__iff__inj, axiom, (! [N2:$tType,M2:$tType]: ((((euclid925273238_space @ M2) & (euclid925273238_space @ N2)) => (! [F2:(M2 > N2)]: (((((real_Vector_linear @ M2) @ N2) @ F2) => ((((((image @ N2) @ M2) @ (((linear_adjoint @ M2) @ N2) @ F2)) @ (top_top @ (set @ N2))) = (top_top @ (set @ M2))) = ((((inj_on @ M2) @ N2) @ F2) @ (top_top @ (set @ M2))))))))))).
% 3.76/1.96 thf(fact_212_adjoint__adjoint, axiom, (! [M2:$tType,N2:$tType]: ((((euclid925273238_space @ N2) & (euclid925273238_space @ M2)) => (! [F2:(N2 > M2)]: (((((real_Vector_linear @ N2) @ M2) @ F2) => ((((linear_adjoint @ M2) @ N2) @ (((linear_adjoint @ N2) @ M2) @ F2)) = F2)))))))).
% 3.76/1.96 thf(fact_213_adjoint__linear, axiom, (! [M2:$tType,N2:$tType]: ((((euclid925273238_space @ N2) & (euclid925273238_space @ M2)) => (! [F2:(N2 > M2)]: (((((real_Vector_linear @ N2) @ M2) @ F2) => (((real_Vector_linear @ M2) @ N2) @ (((linear_adjoint @ N2) @ M2) @ F2))))))))).
% 3.76/1.96 thf(fact_214_linear__surj__adj__imp__inj, axiom, (! [N2:$tType,M2:$tType]: ((((euclid925273238_space @ M2) & (euclid925273238_space @ N2)) => (! [F2:(M2 > N2)]: (((((real_Vector_linear @ M2) @ N2) @ F2) => ((((((image @ N2) @ M2) @ (((linear_adjoint @ M2) @ N2) @ F2)) @ (top_top @ (set @ N2))) = (top_top @ (set @ M2))) => ((((inj_on @ M2) @ N2) @ F2) @ (top_top @ (set @ M2))))))))))).
% 3.76/1.96 thf(fact_215_coplanar__linear__image__eq, axiom, (! [B:$tType,A:$tType]: ((((euclid925273238_space @ A) & (euclid925273238_space @ B)) => (! [F2:(A > B),S:(set @ A)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((coplanar @ B) @ ((((image @ A) @ B) @ F2) @ S)) = ((coplanar @ A) @ S)))))))))).
% 3.76/1.96 thf(fact_216_linear__singular__into__hyperplane, axiom, (! [N2:$tType]: (((euclid925273238_space @ N2) => (! [F2:(N2 > N2)]: (((((real_Vector_linear @ N2) @ N2) @ F2) => ((~ (((((inj_on @ N2) @ N2) @ F2) @ (top_top @ (set @ N2))))) = (? [A7:N2]: (((A7 != (zero_zero @ N2)) & (! [X:N2]: (((((inner_780170721_inner @ N2) @ A7) @ (F2 @ X)) = (zero_zero @ real))))))))))))))).
% 3.76/1.96 thf(fact_217_adjoint__unique, axiom, (! [A:$tType,B:$tType]: ((((inner_real_inner @ B) & (inner_real_inner @ A)) => (! [F2:(A > B),G2:(B > A)]: (((! [X4:A,Y2:B]: (((((inner_780170721_inner @ B) @ (F2 @ X4)) @ Y2) = (((inner_780170721_inner @ A) @ X4) @ (G2 @ Y2))))) => ((((linear_adjoint @ A) @ B) @ F2) = G2)))))))).
% 3.76/1.96 thf(fact_218_hyperplane__eq__Ex, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [A2:A,B2:real]: (((A2 != (zero_zero @ A)) => (~ ((! [X4:A]: (((((inner_780170721_inner @ A) @ A2) @ X4) != B2)))))))))))).
% 3.76/1.96 thf(fact_219_adjoint__works, axiom, (! [N2:$tType,M2:$tType]: ((((euclid925273238_space @ M2) & (euclid925273238_space @ N2)) => (! [F2:(N2 > M2),X2:N2,Y:M2]: (((((real_Vector_linear @ N2) @ M2) @ F2) => ((((inner_780170721_inner @ N2) @ X2) @ ((((linear_adjoint @ N2) @ M2) @ F2) @ Y)) = (((inner_780170721_inner @ M2) @ (F2 @ X2)) @ Y))))))))).
% 3.76/1.96 thf(fact_220_adjoint__clauses_I2_J, axiom, (! [M2:$tType,N2:$tType]: ((((euclid925273238_space @ N2) & (euclid925273238_space @ M2)) => (! [F2:(N2 > M2),Y:M2,X2:N2]: (((((real_Vector_linear @ N2) @ M2) @ F2) => ((((inner_780170721_inner @ N2) @ ((((linear_adjoint @ N2) @ M2) @ F2) @ Y)) @ X2) = (((inner_780170721_inner @ M2) @ Y) @ (F2 @ X2)))))))))).
% 3.76/1.96 thf(fact_221_vector__eq, axiom, (! [A:$tType]: (((inner_real_inner @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [X:A,Y3:A]: ((((((inner_780170721_inner @ A) @ X) @ X) = (((inner_780170721_inner @ A) @ X) @ Y3)) & ((((inner_780170721_inner @ A) @ Y3) @ Y3) = (((inner_780170721_inner @ A) @ X) @ X)))))))))).
% 3.76/1.96 thf(fact_222_vector__eq__ldot, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [Y:A,Z3:A]: (((! [X:A]: (((((inner_780170721_inner @ A) @ X) @ Y) = (((inner_780170721_inner @ A) @ X) @ Z3)))) = (Y = Z3)))))))).
% 3.76/1.96 thf(fact_223_vector__eq__rdot, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X2:A,Y:A]: (((! [Z2:A]: (((((inner_780170721_inner @ A) @ X2) @ Z2) = (((inner_780170721_inner @ A) @ Y) @ Z2)))) = (X2 = Y)))))))).
% 3.76/1.96 thf(fact_224_coplanar__linear__image, axiom, (! [B:$tType,A:$tType]: ((((euclid925273238_space @ A) & (real_V55928688vector @ B)) => (! [S:(set @ A),F2:(A > B)]: ((((coplanar @ A) @ S) => ((((real_Vector_linear @ A) @ B) @ F2) => ((coplanar @ B) @ ((((image @ A) @ B) @ F2) @ S)))))))))).
% 3.76/1.96 thf(fact_225_inner__eq__zero__iff, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X2:A]: ((((((inner_780170721_inner @ A) @ X2) @ X2) = (zero_zero @ real)) = (X2 = (zero_zero @ A))))))))).
% 3.76/1.96 thf(fact_226_inner__zero__right, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X2:A]: (((((inner_780170721_inner @ A) @ X2) @ (zero_zero @ A)) = (zero_zero @ real)))))))).
% 3.76/1.96 thf(fact_227_all__zero__iff, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X2:A]: (((! [U:A]: (((((inner_780170721_inner @ A) @ X2) @ U) = (zero_zero @ real)))) = (X2 = (zero_zero @ A))))))))).
% 3.76/1.96 thf(fact_228_inner__zero__left, axiom, (! [A:$tType]: (((inner_real_inner @ A) => (! [X2:A]: (((((inner_780170721_inner @ A) @ (zero_zero @ A)) @ X2) = (zero_zero @ real)))))))).
% 3.76/1.96 thf(fact_229_aff__dim__injective__linear__image, axiom, (! [B:$tType,A:$tType]: ((((euclid925273238_space @ A) & (euclid925273238_space @ B)) => (! [F2:(A > B),S:(set @ A)]: (((((real_Vector_linear @ A) @ B) @ F2) => (((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((aff_dim @ B) @ ((((image @ A) @ B) @ F2) @ S)) = ((aff_dim @ A) @ S)))))))))).
% 3.76/1.96 thf(fact_230_bij__swap__comp, axiom, (! [B:$tType,A:$tType,P:(A > B),A2:B,B2:B]: (((((((bij_betw @ A) @ B) @ P) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((((comp @ B) @ B) @ A) @ (((((swap @ B) @ B) @ A2) @ B2) @ (id @ B))) @ P) = (((((swap @ A) @ B) @ (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ P) @ A2)) @ (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ P) @ B2)) @ P)))))).
% 3.76/1.96 thf(fact_231_inj__linear__imp__inv__bounded__linear, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [F2:(A > A)]: (((((real_V1632203528linear @ A) @ A) @ F2) => (((((inj_on @ A) @ A) @ F2) @ (top_top @ (set @ A))) => (((real_V1632203528linear @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F2)))))))))).
% 3.76/1.96 thf(fact_232_inv__swap__id, axiom, (! [A:$tType,A2:A,B2:A]: ((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A))) = (((((swap @ A) @ A) @ A2) @ B2) @ (id @ A)))))).
% 3.76/1.96 thf(fact_233_inj__linear__imp__inv__linear, axiom, (! [A:$tType]: (((euclid925273238_space @ A) => (! [F2:(A > A)]: (((((real_Vector_linear @ A) @ A) @ F2) => (((((inj_on @ A) @ A) @ F2) @ (top_top @ (set @ A))) => (((real_Vector_linear @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F2)))))))))).
% 3.76/1.96 thf(fact_234_inv__o__cancel, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((((comp @ B) @ A) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2)) @ F2) = (id @ A)))))).
% 3.76/1.96 thf(fact_235_o__inv__o__cancel, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(A > B),G2:(A > C)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((((comp @ B) @ C) @ A) @ (((((comp @ A) @ C) @ B) @ G2) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2))) @ F2) = G2))))).
% 3.76/1.96 thf(fact_236_inv__into__f__f, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),X2:A]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((((member @ A) @ X2) @ A4) => ((((((hilbert_inv_into @ A) @ B) @ A4) @ F2) @ (F2 @ X2)) = X2)))))).
% 3.76/1.96 thf(fact_237_inv__id, axiom, (! [A:$tType]: ((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ (id @ A)) = (id @ A))))).
% 3.76/1.96 thf(fact_238_inv__equality, axiom, (! [A:$tType,B:$tType,G2:(B > A),F2:(A > B)]: (((! [X4:A]: (((G2 @ (F2 @ X4)) = X4))) => ((! [Y2:B]: (((F2 @ (G2 @ Y2)) = Y2))) => (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2) = G2)))))).
% 3.76/1.96 thf(fact_239_inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),X2:A,Y:B]: ((((((inj_on @ A) @ B) @ F2) @ A4) => ((((member @ A) @ X2) @ A4) => (((F2 @ X2) = Y) => ((((((hilbert_inv_into @ A) @ B) @ A4) @ F2) @ Y) = X2))))))).
% 3.76/1.96 thf(fact_240_bij__betw__inv__into, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),B4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ B4) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ A4) @ F2)) @ B4) @ A4))))).
% 3.76/1.96 thf(fact_241_inv__into__inv__into__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),A8:(set @ B),A2:A]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ A8) => ((((member @ A) @ A2) @ A4) => ((((((hilbert_inv_into @ B) @ A) @ A8) @ ((((hilbert_inv_into @ A) @ B) @ A4) @ F2)) @ A2) = (F2 @ A2))))))).
% 3.76/1.96 thf(fact_242_bij__betw__inv__into__left, axiom, (! [B:$tType,A:$tType,F2:(A > B),A4:(set @ A),A8:(set @ B),A2:A]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ A8) => ((((member @ A) @ A2) @ A4) => ((((((hilbert_inv_into @ A) @ B) @ A4) @ F2) @ (F2 @ A2)) = A2)))))).
% 3.76/1.96 thf(fact_243_bij__betw__inv__into__right, axiom, (! [A:$tType,B:$tType,F2:(A > B),A4:(set @ A),A8:(set @ B),A10:B]: (((((((bij_betw @ A) @ B) @ F2) @ A4) @ A8) => ((((member @ B) @ A10) @ A8) => ((F2 @ (((((hilbert_inv_into @ A) @ B) @ A4) @ F2) @ A10)) = A10)))))).
% 3.76/1.96 thf(fact_244_surj__f__inv__f, axiom, (! [B:$tType,A:$tType,F2:(B > A),Y:A]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((F2 @ (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F2) @ Y)) = Y))))).
% 3.76/1.96 thf(fact_245_surj__iff__all, axiom, (! [B:$tType,A:$tType,F2:(B > A)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [X:A]: (((F2 @ (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F2) @ X)) = X))))))).
% 3.76/1.96 thf(fact_246_image__f__inv__f, axiom, (! [B:$tType,A:$tType,F2:(B > A),A4:(set @ A)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ F2) @ ((((image @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F2)) @ A4)) = A4))))).
% 3.76/1.96 thf(fact_247_surj__imp__inv__eq, axiom, (! [B:$tType,A:$tType,F2:(B > A),G2:(A > B)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((! [X4:B]: (((G2 @ (F2 @ X4)) = X4))) => (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F2) = G2)))))).
% 3.76/1.96 thf(fact_248_inv__f__f, axiom, (! [B:$tType,A:$tType,F2:(A > B),X2:A]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2) @ (F2 @ X2)) = X2))))).
% 3.76/1.96 thf(fact_249_inv__f__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),X2:A,Y:B]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((F2 @ X2) = Y) => ((((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2) @ Y) = X2)))))).
% 3.76/1.96 thf(fact_250_inj__imp__inv__eq, axiom, (! [A:$tType,B:$tType,F2:(A > B),G2:(B > A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((! [X4:B]: (((F2 @ (G2 @ X4)) = X4))) => (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2) = G2)))))).
% 3.76/1.96 thf(fact_251_bij__imp__bij__inv, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2)) @ (top_top @ (set @ B))) @ (top_top @ (set @ A))))))).
% 3.76/1.96 thf(fact_252_bij__inv__eq__iff, axiom, (! [A:$tType,B:$tType,P:(A > B),X2:A,Y:B]: (((((((bij_betw @ A) @ B) @ P) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((X2 = (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ P) @ Y)) = ((P @ X2) = Y)))))).
% 3.76/1.96 thf(fact_253_inv__inv__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F2)) = F2))))).
% 3.76/1.96 thf(fact_254_surj__imp__inj__inv, axiom, (! [B:$tType,A:$tType,F2:(B > A)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((inj_on @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F2)) @ (top_top @ (set @ A))))))).
% 3.76/1.96 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___HOL_Otype, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (type @ A))))).
% 3.76/1.96 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (zero @ A))))).
% 3.76/1.96 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Inner__Product_Oreal__inner, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (inner_real_inner @ A))))).
% 3.76/1.96 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Oreal__vector, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (real_V1076094709vector @ A))))).
% 3.76/1.96 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Euclidean__Space_Oeuclidean__space, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (euclid925273238_space @ A))))).
% 3.76/1.96 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Topological__Spaces_Otopological__space, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (topolo503727757_space @ A))))).
% 3.76/1.96 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Oreal__normed__vector, axiom, (! [A:$tType]: (((execut510477386_space @ A) => (real_V55928688vector @ A))))).
% 3.76/1.96 thf(tcon_fun___Topological__Spaces_Otopological__space, axiom, (! [A11:$tType,A12:$tType]: (((topolo503727757_space @ A12) => (topolo503727757_space @ (A11 > A12)))))).
% 3.76/1.96 thf(tcon_fun___Orderings_Otop, axiom, (! [A11:$tType,A12:$tType]: (((top @ A12) => (top @ (A11 > A12)))))).
% 3.76/1.96 thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_1, axiom, (topolo503727757_space @ int)).
% 3.76/1.96 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 3.76/1.96 thf(tcon_Set_Oset___Orderings_Otop_2, axiom, (! [A11:$tType]: ((top @ (set @ A11))))).
% 3.76/1.96 thf(tcon_Set_Oset___Groups_Ozero_3, axiom, (! [A11:$tType]: (((zero @ A11) => (zero @ (set @ A11)))))).
% 3.76/1.96 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_4, axiom, (topolo503727757_space @ $o)).
% 3.76/1.96 thf(tcon_HOL_Obool___Orderings_Otop_5, axiom, (top @ $o)).
% 3.76/1.96 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space, axiom, (execut510477386_space @ real)).
% 3.76/1.96 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector, axiom, (real_V55928688vector @ real)).
% 3.76/1.96 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_6, axiom, (topolo503727757_space @ real)).
% 3.76/1.96 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space, axiom, (euclid925273238_space @ real)).
% 3.76/1.96 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector, axiom, (real_V1076094709vector @ real)).
% 3.76/1.96 thf(tcon_Real_Oreal___Inner__Product_Oreal__inner, axiom, (inner_real_inner @ real)).
% 3.76/1.96 thf(tcon_Real_Oreal___Groups_Ozero_7, axiom, (zero @ real)).
% 3.76/1.96 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_8, axiom, (real_V55928688vector @ complex)).
% 3.76/1.96 thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_9, axiom, (topolo503727757_space @ complex)).
% 3.76/1.96 thf(tcon_Complex_Ocomplex___Euclidean__Space_Oeuclidean__space_10, axiom, (euclid925273238_space @ complex)).
% 3.76/1.96 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_11, axiom, (real_V1076094709vector @ complex)).
% 3.76/1.96 thf(tcon_Complex_Ocomplex___Inner__Product_Oreal__inner_12, axiom, (inner_real_inner @ complex)).
% 3.76/1.96 thf(tcon_Complex_Ocomplex___Groups_Ozero_13, axiom, (zero @ complex)).
% 3.76/1.96 thf(tfree_0, hypothesis, (execut510477386_space @ a)).
% 3.76/1.96 thf(conj_0, conjecture, (((path_pathfinish @ complex) @ (((((comp @ a) @ complex) @ real) @ (poinca1531302022lex_of @ a)) @ c)) = ((path_pathstart @ complex) @ (((((comp @ a) @ complex) @ real) @ (poinca1531302022lex_of @ a)) @ c)))).
% 3.76/1.96 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2DTF11658.p
% 3.76/1.96 ------------------------
% 3.76/2.02 ---- Cleaned THF ---
% 3.76/2.02 thf(ty_t_Bounded__Linear__Function_Oblinfun,type,
% 3.76/2.02 bounde2145540817linfun: $tType > $tType > $tType ).
% 3.76/2.02
% 3.76/2.02 thf(ty_t_Complex_Ocomplex,type,
% 3.76/2.02 complex: $tType ).
% 3.76/2.02
% 3.76/2.02 thf(ty_t_Real_Oreal,type,
% 3.76/2.02 real: $tType ).
% 3.76/2.02
% 3.76/2.02 thf(ty_t_Set_Oset,type,
% 3.76/2.02 set: $tType > $tType ).
% 3.76/2.02
% 3.76/2.02 thf(ty_t_Int_Oint,type,
% 3.76/2.02 int: $tType ).
% 3.76/2.02
% 3.76/2.02 thf(ty_tf_a,type,
% 3.76/2.02 a: $tType ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_HOL_Otype,type,
% 3.76/2.02 type:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_Groups_Ozero,type,
% 3.76/2.02 zero:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_Orderings_Otop,type,
% 3.76/2.02 top:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_Inner__Product_Oreal__inner,type,
% 3.76/2.02 inner_real_inner:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
% 3.76/2.02 real_V1076094709vector:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
% 3.76/2.02 euclid925273238_space:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 3.76/2.02 topolo503727757_space:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 3.76/2.02 real_V55928688vector:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
% 3.76/2.02 execut510477386_space:
% 3.76/2.02 !>[A: $tType] : $o ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Affine_Oaff__dim,type,
% 3.76/2.02 aff_dim:
% 3.76/2.02 !>[A: $tType] : ( ( set @ A ) > int ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Fun_Obij__betw,type,
% 3.76/2.02 bij_betw:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Fun_Ocomp,type,
% 3.76/2.02 comp:
% 3.76/2.02 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Fun_Oid,type,
% 3.76/2.02 id:
% 3.76/2.02 !>[A: $tType] : ( A > A ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Fun_Oinj__on,type,
% 3.76/2.02 inj_on:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Fun_Oswap,type,
% 3.76/2.02 swap:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Fun_Othe__inv__into,type,
% 3.76/2.02 the_inv_into:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.76/2.02 zero_zero:
% 3.76/2.02 !>[A: $tType] : A ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 3.76/2.02 hilbert_inv_into:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner,type,
% 3.76/2.02 inner_780170721_inner:
% 3.76/2.02 !>[A: $tType] : ( A > A > real ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Linear__Algebra_Oadjoint,type,
% 3.76/2.02 linear_adjoint:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( A > B ) > B > A ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.76/2.02 top_top:
% 3.76/2.02 !>[A: $tType] : A ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Path__Connected_Oarc,type,
% 3.76/2.02 path_arc:
% 3.76/2.02 !>[A: $tType] : ( ( real > A ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Path__Connected_Opath,type,
% 3.76/2.02 path_path:
% 3.76/2.02 !>[A: $tType] : ( ( real > A ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Path__Connected_Opathfinish,type,
% 3.76/2.02 path_pathfinish:
% 3.76/2.02 !>[A: $tType] : ( ( real > A ) > A ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Path__Connected_Opathstart,type,
% 3.76/2.02 path_pathstart:
% 3.76/2.02 !>[A: $tType] : ( ( real > A ) > A ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Path__Connected_Orectpath,type,
% 3.76/2.02 path_rectpath: complex > complex > real > complex ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Path__Connected_Osimple__path,type,
% 3.76/2.02 path_simple_path:
% 3.76/2.02 !>[A: $tType] : ( ( real > A ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2,type,
% 3.76/2.02 poinca1076805489pen_R2:
% 3.76/2.02 !>[A: $tType] : ( ( A > A ) > ( A > ( bounde2145540817linfun @ A @ A ) ) > ( set @ A ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Ocomplex__of,type,
% 3.76/2.02 poinca1531302022lex_of:
% 3.76/2.02 !>[A: $tType] : ( A > complex ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Oreal__of,type,
% 3.76/2.02 poinca179318102eal_of:
% 3.76/2.02 !>[A: $tType] : ( complex > A ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
% 3.76/2.02 real_V1632203528linear:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
% 3.76/2.02 real_V1072672427axioms:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Real__Vector__Spaces_Olinear,type,
% 3.76/2.02 real_Vector_linear:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist,type,
% 3.76/2.02 iso_tu2011167877assist:
% 3.76/2.02 !>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > A > ( B > B ) > A > B > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Set_OCollect,type,
% 3.76/2.02 collect:
% 3.76/2.02 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Set_Oimage,type,
% 3.76/2.02 image:
% 3.76/2.02 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_Starlike_Ocoplanar,type,
% 3.76/2.02 coplanar:
% 3.76/2.02 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_c_member,type,
% 3.76/2.02 member:
% 3.76/2.02 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.76/2.02
% 3.76/2.02 thf(sy_v_c,type,
% 3.76/2.02 c: real > a ).
% 3.76/2.02
% 3.76/2.02 thf(fact_0_assms_I2_J,axiom,
% 3.76/2.02 ( ( path_pathfinish @ a @ c )
% 3.76/2.02 = ( path_pathstart @ a @ c ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_1_assms_I1_J,axiom,
% 3.76/2.02 path_simple_path @ a @ c ).
% 3.76/2.02
% 3.76/2.02 thf(fact_2_a1,axiom,
% 3.76/2.02 path_simple_path @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_3_comp__apply,axiom,
% 3.76/2.02 ! [C: $tType,A: $tType,B: $tType] :
% 3.76/2.02 ( ( comp @ B @ A @ C )
% 3.76/2.02 = ( ^ [F: B > A,G: C > B,X: C] : ( F @ ( G @ X ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_4_pathfinish__compose,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType] :
% 3.76/2.02 ( ( ( topolo503727757_space @ B )
% 3.76/2.02 & ( topolo503727757_space @ A ) )
% 3.76/2.02 => ! [F2: B > A,P: real > B] :
% 3.76/2.02 ( ( path_pathfinish @ A @ ( comp @ B @ A @ real @ F2 @ P ) )
% 3.76/2.02 = ( F2 @ ( path_pathfinish @ B @ P ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_5_pathstart__compose,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType] :
% 3.76/2.02 ( ( ( topolo503727757_space @ B )
% 3.76/2.02 & ( topolo503727757_space @ A ) )
% 3.76/2.02 => ! [F2: B > A,P: real > B] :
% 3.76/2.02 ( ( path_pathstart @ A @ ( comp @ B @ A @ real @ F2 @ P ) )
% 3.76/2.02 = ( F2 @ ( path_pathstart @ B @ P ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_6_complex__of__bounded__linear,axiom,
% 3.76/2.02 real_V1632203528linear @ a @ complex @ ( poinca1531302022lex_of @ a ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_7_complex__of__linear,axiom,
% 3.76/2.02 real_Vector_linear @ a @ complex @ ( poinca1531302022lex_of @ a ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_8_comp__def,axiom,
% 3.76/2.02 ! [A: $tType,C: $tType,B: $tType] :
% 3.76/2.02 ( ( comp @ B @ C @ A )
% 3.76/2.02 = ( ^ [F: B > C,G: A > B,X: A] : ( F @ ( G @ X ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_9_comp__assoc,axiom,
% 3.76/2.02 ! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H: A > C] :
% 3.76/2.02 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H )
% 3.76/2.02 = ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_10_comp__eq__dest,axiom,
% 3.76/2.02 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
% 3.76/2.02 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 3.76/2.02 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 3.76/2.02 => ( ( A2 @ ( B2 @ V ) )
% 3.76/2.02 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_11_comp__eq__elim,axiom,
% 3.76/2.02 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
% 3.76/2.02 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 3.76/2.02 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 3.76/2.02 => ! [V2: A] :
% 3.76/2.02 ( ( A2 @ ( B2 @ V2 ) )
% 3.76/2.02 = ( C2 @ ( D2 @ V2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_12_comp__cong,axiom,
% 3.76/2.02 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F2: B > A,G2: C > B,X2: C,F3: D > A,G3: E > D,X3: E] :
% 3.76/2.02 ( ( ( F2 @ ( G2 @ X2 ) )
% 3.76/2.02 = ( F3 @ ( G3 @ X3 ) ) )
% 3.76/2.02 => ( ( comp @ B @ A @ C @ F2 @ G2 @ X2 )
% 3.76/2.02 = ( comp @ D @ A @ E @ F3 @ G3 @ X3 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_13_comp__eq__dest__lhs,axiom,
% 3.76/2.02 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 3.76/2.02 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 3.76/2.02 = C2 )
% 3.76/2.02 => ( ( A2 @ ( B2 @ V ) )
% 3.76/2.02 = ( C2 @ V ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_14_pathstart__linear__image__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V1076094709vector @ A )
% 3.76/2.02 & ( topolo503727757_space @ A )
% 3.76/2.02 & ( real_V1076094709vector @ B )
% 3.76/2.02 & ( topolo503727757_space @ B ) )
% 3.76/2.02 => ! [F2: A > B,G2: real > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( path_pathstart @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 3.76/2.02 = ( F2 @ ( path_pathstart @ A @ G2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_15_pathfinish__linear__image,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V1076094709vector @ A )
% 3.76/2.02 & ( topolo503727757_space @ A )
% 3.76/2.02 & ( real_V1076094709vector @ B )
% 3.76/2.02 & ( topolo503727757_space @ B ) )
% 3.76/2.02 => ! [F2: A > B,G2: real > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( path_pathfinish @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 3.76/2.02 = ( F2 @ ( path_pathfinish @ A @ G2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_16_bounded__linear_Olinear,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V55928688vector @ A )
% 3.76/2.02 & ( real_V55928688vector @ B ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_V1632203528linear @ A @ B @ F2 )
% 3.76/2.02 => ( real_Vector_linear @ A @ B @ F2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_17_linear__conv__bounded__linear,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ A )
% 3.76/2.02 & ( real_V55928688vector @ B ) )
% 3.76/2.02 => ( ( real_Vector_linear @ A @ B )
% 3.76/2.02 = ( real_V1632203528linear @ A @ B ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_18_linear__compose,axiom,
% 3.76/2.02 ! [C: $tType,B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V1076094709vector @ A )
% 3.76/2.02 & ( real_V1076094709vector @ B )
% 3.76/2.02 & ( real_V1076094709vector @ C ) )
% 3.76/2.02 => ! [F2: A > B,G2: B > C] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( real_Vector_linear @ B @ C @ G2 )
% 3.76/2.02 => ( real_Vector_linear @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_19__092_060open_062_092_060And_062g_O_Ainj_Acomplex__of_A_092_060Longrightarrow_062_Asimple__path_A_Icomplex__of_A_092_060circ_062_Ag_J_A_061_Asimple__path_Ag_092_060close_062,axiom,
% 3.76/2.02 ! [G2: real > a] :
% 3.76/2.02 ( ( inj_on @ a @ complex @ ( poinca1531302022lex_of @ a ) @ ( top_top @ ( set @ a ) ) )
% 3.76/2.02 => ( ( path_simple_path @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ G2 ) )
% 3.76/2.02 = ( path_simple_path @ a @ G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_20_pathfinish__rectpath,axiom,
% 3.76/2.02 ! [A1: complex,A3: complex] :
% 3.76/2.02 ( ( path_pathfinish @ complex @ ( path_rectpath @ A1 @ A3 ) )
% 3.76/2.02 = A1 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_21_pathstart__rectpath,axiom,
% 3.76/2.02 ! [A1: complex,A3: complex] :
% 3.76/2.02 ( ( path_pathstart @ complex @ ( path_rectpath @ A1 @ A3 ) )
% 3.76/2.02 = A1 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_22_bounded__linear_Obounded__linear,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V55928688vector @ A )
% 3.76/2.02 & ( real_V55928688vector @ B ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_V1632203528linear @ A @ B @ F2 )
% 3.76/2.02 => ( real_V1632203528linear @ A @ B @ F2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_23_rewriteR__comp__comp2,axiom,
% 3.76/2.02 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R2: A > D,F2: B > E,L: D > E] :
% 3.76/2.02 ( ( ( comp @ C @ B @ A @ G2 @ H )
% 3.76/2.02 = ( comp @ D @ B @ A @ R1 @ R2 ) )
% 3.76/2.02 => ( ( ( comp @ B @ E @ D @ F2 @ R1 )
% 3.76/2.02 = L )
% 3.76/2.02 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F2 @ G2 ) @ H )
% 3.76/2.02 = ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_24_rewriteL__comp__comp2,axiom,
% 3.76/2.02 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F2: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 3.76/2.02 ( ( ( comp @ C @ B @ A @ F2 @ G2 )
% 3.76/2.02 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 3.76/2.02 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 3.76/2.02 = R )
% 3.76/2.02 => ( ( comp @ C @ B @ E @ F2 @ ( comp @ A @ C @ E @ G2 @ H ) )
% 3.76/2.02 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_25_rewriteR__comp__comp,axiom,
% 3.76/2.02 ! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R: A > B,F2: B > D] :
% 3.76/2.02 ( ( ( comp @ C @ B @ A @ G2 @ H )
% 3.76/2.02 = R )
% 3.76/2.02 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F2 @ G2 ) @ H )
% 3.76/2.02 = ( comp @ B @ D @ A @ F2 @ R ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_26_rewriteL__comp__comp,axiom,
% 3.76/2.02 ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: C > B,G2: A > C,L: A > B,H: D > A] :
% 3.76/2.02 ( ( ( comp @ C @ B @ A @ F2 @ G2 )
% 3.76/2.02 = L )
% 3.76/2.02 => ( ( comp @ C @ B @ D @ F2 @ ( comp @ A @ C @ D @ G2 @ H ) )
% 3.76/2.02 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_27_injD,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,X2: A,Y: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( ( F2 @ X2 )
% 3.76/2.02 = ( F2 @ Y ) )
% 3.76/2.02 => ( X2 = Y ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_28_injI,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B] :
% 3.76/2.02 ( ! [X4: A,Y2: A] :
% 3.76/2.02 ( ( ( F2 @ X4 )
% 3.76/2.02 = ( F2 @ Y2 ) )
% 3.76/2.02 => ( X4 = Y2 ) )
% 3.76/2.02 => ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_29_inj__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,X2: A,Y: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( ( F2 @ X2 )
% 3.76/2.02 = ( F2 @ Y ) )
% 3.76/2.02 = ( X2 = Y ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_30_inj__def,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( ! [X: A,Y3: A] :
% 3.76/2.02 ( ( ( F2 @ X )
% 3.76/2.02 = ( F2 @ Y3 ) )
% 3.76/2.02 => ( X = Y3 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_31_inj__onD,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( ( F2 @ X2 )
% 3.76/2.02 = ( F2 @ Y ) )
% 3.76/2.02 => ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ Y @ A4 )
% 3.76/2.02 => ( X2 = Y ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_32_inj__onI,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
% 3.76/2.02 ( ! [X4: A,Y2: A] :
% 3.76/2.02 ( ( member @ A @ X4 @ A4 )
% 3.76/2.02 => ( ( member @ A @ Y2 @ A4 )
% 3.76/2.02 => ( ( ( F2 @ X4 )
% 3.76/2.02 = ( F2 @ Y2 ) )
% 3.76/2.02 => ( X4 = Y2 ) ) ) )
% 3.76/2.02 => ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_33_inj__on__def,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( inj_on @ A @ B )
% 3.76/2.02 = ( ^ [F: A > B,A5: set @ A] :
% 3.76/2.02 ! [X: A] :
% 3.76/2.02 ( ( member @ A @ X @ A5 )
% 3.76/2.02 => ! [Y3: A] :
% 3.76/2.02 ( ( member @ A @ Y3 @ A5 )
% 3.76/2.02 => ( ( ( F @ X )
% 3.76/2.02 = ( F @ Y3 ) )
% 3.76/2.02 => ( X = Y3 ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_34_inj__on__cong,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A4: set @ A,F2: A > B,G2: A > B] :
% 3.76/2.02 ( ! [A6: A] :
% 3.76/2.02 ( ( member @ A @ A6 @ A4 )
% 3.76/2.02 => ( ( F2 @ A6 )
% 3.76/2.02 = ( G2 @ A6 ) ) )
% 3.76/2.02 => ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 = ( inj_on @ A @ B @ G2 @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_35_inj__on__eq__iff,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ Y @ A4 )
% 3.76/2.02 => ( ( ( F2 @ X2 )
% 3.76/2.02 = ( F2 @ Y ) )
% 3.76/2.02 = ( X2 = Y ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_36_inj__on__contraD,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( X2 != Y )
% 3.76/2.02 => ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ Y @ A4 )
% 3.76/2.02 => ( ( F2 @ X2 )
% 3.76/2.02 != ( F2 @ Y ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_37_inj__on__inverseI,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A4: set @ A,G2: B > A,F2: A > B] :
% 3.76/2.02 ( ! [X4: A] :
% 3.76/2.02 ( ( member @ A @ X4 @ A4 )
% 3.76/2.02 => ( ( G2 @ ( F2 @ X4 ) )
% 3.76/2.02 = X4 ) )
% 3.76/2.02 => ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_38_inj__compose,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: C > A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( inj_on @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) )
% 3.76/2.02 => ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F2 @ G2 ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_39_inj__on__imageI2,axiom,
% 3.76/2.02 ! [B: $tType,C: $tType,A: $tType,F3: C > B,F2: A > C,A4: set @ A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F3 @ F2 ) @ A4 )
% 3.76/2.02 => ( inj_on @ A @ C @ F2 @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_40_simple__path__linear__image__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ A )
% 3.76/2.02 & ( euclid925273238_space @ B ) )
% 3.76/2.02 => ! [F2: A > B,G2: real > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( path_simple_path @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 3.76/2.02 = ( path_simple_path @ A @ G2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_41_fun_Oinj__map,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,D: $tType,F2: A > B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F2 ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_42_real__of__bounded__linear,axiom,
% 3.76/2.02 real_V1632203528linear @ complex @ a @ ( poinca179318102eal_of @ a ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_43_real__of__linear,axiom,
% 3.76/2.02 real_Vector_linear @ complex @ a @ ( poinca179318102eal_of @ a ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_44_iso__tuple__UNIV__I,axiom,
% 3.76/2.02 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_45_mem__Collect__eq,axiom,
% 3.76/2.02 ! [A: $tType,A2: A,P2: A > $o] :
% 3.76/2.02 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 3.76/2.02 = ( P2 @ A2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_46_Collect__mem__eq,axiom,
% 3.76/2.02 ! [A: $tType,A4: set @ A] :
% 3.76/2.02 ( ( collect @ A
% 3.76/2.02 @ ^ [X: A] : ( member @ A @ X @ A4 ) )
% 3.76/2.02 = A4 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_47_Collect__cong,axiom,
% 3.76/2.02 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.76/2.02 ( ! [X4: A] :
% 3.76/2.02 ( ( P2 @ X4 )
% 3.76/2.02 = ( Q @ X4 ) )
% 3.76/2.02 => ( ( collect @ A @ P2 )
% 3.76/2.02 = ( collect @ A @ Q ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_48_ext,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 3.76/2.02 ( ! [X4: A] :
% 3.76/2.02 ( ( F2 @ X4 )
% 3.76/2.02 = ( G2 @ X4 ) )
% 3.76/2.02 => ( F2 = G2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_49_UNIV__I,axiom,
% 3.76/2.02 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_50_top__apply,axiom,
% 3.76/2.02 ! [C: $tType,D: $tType] :
% 3.76/2.02 ( ( top @ C )
% 3.76/2.02 => ( ( top_top @ ( D > C ) )
% 3.76/2.02 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_51_bounded__linear_Ointro,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V55928688vector @ A )
% 3.76/2.02 & ( real_V55928688vector @ B ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( real_V1072672427axioms @ A @ B @ F2 )
% 3.76/2.02 => ( real_V1632203528linear @ A @ B @ F2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_52_bounded__linear__def,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V55928688vector @ A )
% 3.76/2.02 & ( real_V55928688vector @ B ) )
% 3.76/2.02 => ( ( real_V1632203528linear @ A @ B )
% 3.76/2.02 = ( ^ [F: A > B] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F )
% 3.76/2.02 & ( real_V1072672427axioms @ A @ B @ F ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_53_arc__linear__image__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ A )
% 3.76/2.02 & ( euclid925273238_space @ B ) )
% 3.76/2.02 => ! [F2: A > B,G2: real > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( path_arc @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 3.76/2.02 = ( path_arc @ A @ G2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_54_real__of__inj,axiom,
% 3.76/2.02 inj_on @ complex @ a @ ( poinca179318102eal_of @ a ) @ ( top_top @ ( set @ complex ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_55_top__set__def,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( top_top @ ( set @ A ) )
% 3.76/2.02 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_56_arc__imp__simple__path,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ! [G2: real > A] :
% 3.76/2.02 ( ( path_arc @ A @ G2 )
% 3.76/2.02 => ( path_simple_path @ A @ G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_57_arc__distinct__ends,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ! [G2: real > A] :
% 3.76/2.02 ( ( path_arc @ A @ G2 )
% 3.76/2.02 => ( ( path_pathfinish @ A @ G2 )
% 3.76/2.02 != ( path_pathstart @ A @ G2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_58_bounded__linear_Oaxioms_I2_J,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V55928688vector @ A )
% 3.76/2.02 & ( real_V55928688vector @ B ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_V1632203528linear @ A @ B @ F2 )
% 3.76/2.02 => ( real_V1072672427axioms @ A @ B @ F2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_59_UNIV__eq__I,axiom,
% 3.76/2.02 ! [A: $tType,A4: set @ A] :
% 3.76/2.02 ( ! [X4: A] : ( member @ A @ X4 @ A4 )
% 3.76/2.02 => ( ( top_top @ ( set @ A ) )
% 3.76/2.02 = A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_60_UNIV__witness,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_61_simple__path__imp__arc,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ! [G2: real > A] :
% 3.76/2.02 ( ( path_simple_path @ A @ G2 )
% 3.76/2.02 => ( ( ( path_pathfinish @ A @ G2 )
% 3.76/2.02 != ( path_pathstart @ A @ G2 ) )
% 3.76/2.02 => ( path_arc @ A @ G2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_62_simple__path__eq__arc,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ! [G2: real > A] :
% 3.76/2.02 ( ( ( path_pathfinish @ A @ G2 )
% 3.76/2.02 != ( path_pathstart @ A @ G2 ) )
% 3.76/2.02 => ( ( path_simple_path @ A @ G2 )
% 3.76/2.02 = ( path_arc @ A @ G2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_63_simple__path__cases,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ! [G2: real > A] :
% 3.76/2.02 ( ( path_simple_path @ A @ G2 )
% 3.76/2.02 => ( ( path_arc @ A @ G2 )
% 3.76/2.02 | ( ( path_pathfinish @ A @ G2 )
% 3.76/2.02 = ( path_pathstart @ A @ G2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_64_arc__simple__path,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ( ( path_arc @ A )
% 3.76/2.02 = ( ^ [G: real > A] :
% 3.76/2.02 ( ( path_simple_path @ A @ G )
% 3.76/2.02 & ( ( path_pathfinish @ A @ G )
% 3.76/2.02 != ( path_pathstart @ A @ G ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_65_fun_Omap__comp,axiom,
% 3.76/2.02 ! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
% 3.76/2.02 ( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
% 3.76/2.02 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_66_complex__of__real__of,axiom,
% 3.76/2.02 ( ( comp @ a @ complex @ complex @ ( poinca1531302022lex_of @ a ) @ ( poinca179318102eal_of @ a ) )
% 3.76/2.02 = ( id @ complex ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_67_real__of__complex__of,axiom,
% 3.76/2.02 ( ( comp @ complex @ a @ a @ ( poinca179318102eal_of @ a ) @ ( poinca1531302022lex_of @ a ) )
% 3.76/2.02 = ( id @ a ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_68_real__of__bij,axiom,
% 3.76/2.02 bij_betw @ complex @ a @ ( poinca179318102eal_of @ a ) @ ( top_top @ ( set @ complex ) ) @ ( top_top @ ( set @ a ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_69_complex__of__bij,axiom,
% 3.76/2.02 bij_betw @ a @ complex @ ( poinca1531302022lex_of @ a ) @ ( top_top @ ( set @ a ) ) @ ( top_top @ ( set @ complex ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_70_linear__injective__left__inverse,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V1076094709vector @ A )
% 3.76/2.02 & ( real_V1076094709vector @ B ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ? [G4: B > A] :
% 3.76/2.02 ( ( real_Vector_linear @ B @ A @ G4 )
% 3.76/2.02 & ( ( comp @ B @ A @ A @ G4 @ F2 )
% 3.76/2.02 = ( id @ A ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_71_path__linear__image__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ A )
% 3.76/2.02 & ( euclid925273238_space @ B ) )
% 3.76/2.02 => ! [F2: A > B,G2: real > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( path_path @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
% 3.76/2.02 = ( path_path @ A @ G2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_72_linear__inj__iff__eq__0,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( real_V1076094709vector @ A )
% 3.76/2.02 & ( real_V1076094709vector @ B ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( ! [X: A] :
% 3.76/2.02 ( ( ( F2 @ X )
% 3.76/2.02 = ( zero_zero @ B ) )
% 3.76/2.02 => ( X
% 3.76/2.02 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_73_type__copy__map__cong0,axiom,
% 3.76/2.02 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G2: C > B,X2: C,N: D > A,H: C > D,F2: A > E] :
% 3.76/2.02 ( ( ( M @ ( G2 @ X2 ) )
% 3.76/2.02 = ( N @ ( H @ X2 ) ) )
% 3.76/2.02 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F2 @ M ) @ G2 @ X2 )
% 3.76/2.02 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F2 @ N ) @ H @ X2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_74_id__apply,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( id @ A )
% 3.76/2.02 = ( ^ [X: A] : X ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_75_top1I,axiom,
% 3.76/2.02 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_76_path__rectpath,axiom,
% 3.76/2.02 ! [A2: complex,B2: complex] : ( path_path @ complex @ ( path_rectpath @ A2 @ B2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_77_comp__id,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B] :
% 3.76/2.02 ( ( comp @ A @ B @ A @ F2 @ ( id @ A ) )
% 3.76/2.02 = F2 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_78_id__comp,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,G2: A > B] :
% 3.76/2.02 ( ( comp @ B @ B @ A @ ( id @ B ) @ G2 )
% 3.76/2.02 = G2 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_79_fun_Omap__id,axiom,
% 3.76/2.02 ! [A: $tType,D: $tType,T: D > A] :
% 3.76/2.02 ( ( comp @ A @ A @ D @ ( id @ A ) @ T )
% 3.76/2.02 = T ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_80_bij__betw__id,axiom,
% 3.76/2.02 ! [A: $tType,A4: set @ A] : ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ A4 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_81_bij__id,axiom,
% 3.76/2.02 ! [A: $tType] : ( bij_betw @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_82_DEADID_Oin__rel,axiom,
% 3.76/2.02 ! [B: $tType] :
% 3.76/2.02 ( ( ^ [Y4: B,Z: B] : ( Y4 = Z ) )
% 3.76/2.02 = ( ^ [A7: B,B3: B] :
% 3.76/2.02 ? [Z2: B] :
% 3.76/2.02 ( ( member @ B @ Z2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 & ( ( id @ B @ Z2 )
% 3.76/2.02 = A7 )
% 3.76/2.02 & ( ( id @ B @ Z2 )
% 3.76/2.02 = B3 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_83_top__conj_I1_J,axiom,
% 3.76/2.02 ! [A: $tType,X2: A,P2: $o] :
% 3.76/2.02 ( ( ( top_top @ ( A > $o ) @ X2 )
% 3.76/2.02 & P2 )
% 3.76/2.02 = P2 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_84_top__conj_I2_J,axiom,
% 3.76/2.02 ! [A: $tType,P2: $o,X2: A] :
% 3.76/2.02 ( ( P2
% 3.76/2.02 & ( top_top @ ( A > $o ) @ X2 ) )
% 3.76/2.02 = P2 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_85_fun_Omap__id0,axiom,
% 3.76/2.02 ! [A: $tType,D: $tType] :
% 3.76/2.02 ( ( comp @ A @ A @ D @ ( id @ A ) )
% 3.76/2.02 = ( id @ ( D > A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_86_id__def,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( id @ A )
% 3.76/2.02 = ( ^ [X: A] : X ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_87_bij__betwE,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ! [X5: A] :
% 3.76/2.02 ( ( member @ A @ X5 @ A4 )
% 3.76/2.02 => ( member @ B @ ( F2 @ X5 ) @ B4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_88_eq__id__iff,axiom,
% 3.76/2.02 ! [A: $tType,F2: A > A] :
% 3.76/2.02 ( ( ! [X: A] :
% 3.76/2.02 ( ( F2 @ X )
% 3.76/2.02 = X ) )
% 3.76/2.02 = ( F2
% 3.76/2.02 = ( id @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_89_bij__betw__inv,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ? [G4: B > A] : ( bij_betw @ B @ A @ G4 @ B4 @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_90_bij__betw__cong,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,A4: set @ A,F2: A > B,G2: A > B,A8: set @ B] :
% 3.76/2.02 ( ! [A6: A] :
% 3.76/2.02 ( ( member @ A @ A6 @ A4 )
% 3.76/2.02 => ( ( F2 @ A6 )
% 3.76/2.02 = ( G2 @ A6 ) ) )
% 3.76/2.02 => ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 3.76/2.02 = ( bij_betw @ A @ B @ G2 @ A4 @ A8 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_91_bij__betw__apply,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B,A2: A] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ( ( member @ A @ A2 @ A4 )
% 3.76/2.02 => ( member @ B @ ( F2 @ A2 ) @ B4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_92_bij__betw__id__iff,axiom,
% 3.76/2.02 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 3.76/2.02 ( ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ B4 )
% 3.76/2.02 = ( A4 = B4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_93_bij__betw__iff__bijections,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( bij_betw @ A @ B )
% 3.76/2.02 = ( ^ [F: A > B,A5: set @ A,B5: set @ B] :
% 3.76/2.02 ? [G: B > A] :
% 3.76/2.02 ( ! [X: A] :
% 3.76/2.02 ( ( member @ A @ X @ A5 )
% 3.76/2.02 => ( ( member @ B @ ( F @ X ) @ B5 )
% 3.76/2.02 & ( ( G @ ( F @ X ) )
% 3.76/2.02 = X ) ) )
% 3.76/2.02 & ! [X: B] :
% 3.76/2.02 ( ( member @ B @ X @ B5 )
% 3.76/2.02 => ( ( member @ A @ ( G @ X ) @ A5 )
% 3.76/2.02 & ( ( F @ ( G @ X ) )
% 3.76/2.02 = X ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_94_o__bij,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,G2: B > A,F2: A > B] :
% 3.76/2.02 ( ( ( comp @ B @ A @ A @ G2 @ F2 )
% 3.76/2.02 = ( id @ A ) )
% 3.76/2.02 => ( ( ( comp @ A @ B @ B @ F2 @ G2 )
% 3.76/2.02 = ( id @ B ) )
% 3.76/2.02 => ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_95_bij__pointE,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,Y: B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ~ ! [X4: A] :
% 3.76/2.02 ( ( Y
% 3.76/2.02 = ( F2 @ X4 ) )
% 3.76/2.02 => ~ ! [X6: A] :
% 3.76/2.02 ( ( Y
% 3.76/2.02 = ( F2 @ X6 ) )
% 3.76/2.02 => ( X6 = X4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_96_bij__betw__trans,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,C: $tType,F2: A > B,A4: set @ A,B4: set @ B,G2: B > C,C3: set @ C] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ( ( bij_betw @ B @ C @ G2 @ B4 @ C3 )
% 3.76/2.02 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ A4 @ C3 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_97_bij__betw__comp__iff,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,C: $tType,F2: A > B,A4: set @ A,A8: set @ B,F3: B > C,A9: set @ C] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 3.76/2.02 => ( ( bij_betw @ B @ C @ F3 @ A8 @ A9 )
% 3.76/2.02 = ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F3 @ F2 ) @ A4 @ A9 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_98_bij__betw__imp__inj__on,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_99_pointfree__idE,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: B > A,G2: A > B,X2: A] :
% 3.76/2.02 ( ( ( comp @ B @ A @ A @ F2 @ G2 )
% 3.76/2.02 = ( id @ A ) )
% 3.76/2.02 => ( ( F2 @ ( G2 @ X2 ) )
% 3.76/2.02 = X2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_100_comp__eq__id__dest,axiom,
% 3.76/2.02 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 3.76/2.02 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 3.76/2.02 = ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
% 3.76/2.02 => ( ( A2 @ ( B2 @ V ) )
% 3.76/2.02 = ( C2 @ V ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_101_arc__imp__path,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ! [G2: real > A] :
% 3.76/2.02 ( ( path_arc @ A @ G2 )
% 3.76/2.02 => ( path_path @ A @ G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_102_inj__on__id,axiom,
% 3.76/2.02 ! [A: $tType,A4: set @ A] : ( inj_on @ A @ A @ ( id @ A ) @ A4 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_103_simple__path__imp__path,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ! [G2: real > A] :
% 3.76/2.02 ( ( path_simple_path @ A @ G2 )
% 3.76/2.02 => ( path_path @ A @ G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_104_module__hom__id,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( real_V1076094709vector @ A )
% 3.76/2.02 => ( real_Vector_linear @ A @ A @ ( id @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_105_linear__simps_I3_J,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType] :
% 3.76/2.02 ( ( ( real_V55928688vector @ B )
% 3.76/2.02 & ( real_V55928688vector @ A ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_V1632203528linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( F2 @ ( zero_zero @ A ) )
% 3.76/2.02 = ( zero_zero @ B ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_106_linear__0,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType] :
% 3.76/2.02 ( ( ( real_V1076094709vector @ B )
% 3.76/2.02 & ( real_V1076094709vector @ A ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( F2 @ ( zero_zero @ A ) )
% 3.76/2.02 = ( zero_zero @ B ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_107_pathstart__def,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A )
% 3.76/2.02 => ( ( path_pathstart @ A )
% 3.76/2.02 = ( ^ [G: real > A] : ( G @ ( zero_zero @ real ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_108_bij__comp,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: B > C] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( ( bij_betw @ B @ C @ G2 @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ C ) ) )
% 3.76/2.02 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_109_bij__is__inj,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_110_right__inverse__linear,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( euclid925273238_space @ A )
% 3.76/2.02 => ! [F2: A > A,G2: A > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ A @ F2 )
% 3.76/2.02 => ( ( ( comp @ A @ A @ A @ F2 @ G2 )
% 3.76/2.02 = ( id @ A ) )
% 3.76/2.02 => ( real_Vector_linear @ A @ A @ G2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_111_linear__inverse__left,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( euclid925273238_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ A @ F2 )
% 3.76/2.02 => ( ( real_Vector_linear @ A @ A @ F3 )
% 3.76/2.02 => ( ( ( comp @ A @ A @ A @ F2 @ F3 )
% 3.76/2.02 = ( id @ A ) )
% 3.76/2.02 = ( ( comp @ A @ A @ A @ F3 @ F2 )
% 3.76/2.02 = ( id @ A ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_112_left__inverse__linear,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( euclid925273238_space @ A )
% 3.76/2.02 => ! [F2: A > A,G2: A > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ A @ F2 )
% 3.76/2.02 => ( ( ( comp @ A @ A @ A @ G2 @ F2 )
% 3.76/2.02 = ( id @ A ) )
% 3.76/2.02 => ( real_Vector_linear @ A @ A @ G2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_113_top__empty__eq,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( top_top @ ( A > $o ) )
% 3.76/2.02 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_114_left__right__inverse__eq,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: B > A,G2: A > B,H: B > A] :
% 3.76/2.02 ( ( ( comp @ B @ A @ A @ F2 @ G2 )
% 3.76/2.02 = ( id @ A ) )
% 3.76/2.02 => ( ( ( comp @ A @ B @ B @ G2 @ H )
% 3.76/2.02 = ( id @ B ) )
% 3.76/2.02 => ( F2 = H ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_115_zero__reorient,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( zero @ A )
% 3.76/2.02 => ! [X2: A] :
% 3.76/2.02 ( ( ( zero_zero @ A )
% 3.76/2.02 = X2 )
% 3.76/2.02 = ( X2
% 3.76/2.02 = ( zero_zero @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_116_isomorphism__expand,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: B > A,G2: A > B] :
% 3.76/2.02 ( ( ( ( comp @ B @ A @ A @ F2 @ G2 )
% 3.76/2.02 = ( id @ A ) )
% 3.76/2.02 & ( ( comp @ A @ B @ B @ G2 @ F2 )
% 3.76/2.02 = ( id @ B ) ) )
% 3.76/2.02 = ( ! [X: A] :
% 3.76/2.02 ( ( F2 @ ( G2 @ X ) )
% 3.76/2.02 = X )
% 3.76/2.02 & ! [X: B] :
% 3.76/2.02 ( ( G2 @ ( F2 @ X ) )
% 3.76/2.02 = X ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_117_bijI_H,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B] :
% 3.76/2.02 ( ! [X4: A,Y2: A] :
% 3.76/2.02 ( ( ( F2 @ X4 )
% 3.76/2.02 = ( F2 @ Y2 ) )
% 3.76/2.02 = ( X4 = Y2 ) )
% 3.76/2.02 => ( ! [Y2: B] :
% 3.76/2.02 ? [X5: A] :
% 3.76/2.02 ( Y2
% 3.76/2.02 = ( F2 @ X5 ) )
% 3.76/2.02 => ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_118_bij__iff,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( ! [X: B] :
% 3.76/2.02 ? [Y3: A] :
% 3.76/2.02 ( ( ( F2 @ Y3 )
% 3.76/2.02 = X )
% 3.76/2.02 & ! [Z2: A] :
% 3.76/2.02 ( ( ( F2 @ Z2 )
% 3.76/2.02 = X )
% 3.76/2.02 => ( Z2 = Y3 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_119_c1__on__open__R2_Ocomplex__of__real__of,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( ( comp @ A @ complex @ complex @ ( poinca1531302022lex_of @ A ) @ ( poinca179318102eal_of @ A ) )
% 3.76/2.02 = ( id @ complex ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_120_c1__on__open__R2_Oreal__of__bij,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( bij_betw @ complex @ A @ ( poinca179318102eal_of @ A ) @ ( top_top @ ( set @ complex ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_121_c1__on__open__R2_Oreal__of__bounded__linear,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( real_V1632203528linear @ complex @ A @ ( poinca179318102eal_of @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_122_c1__on__open__R2_Ocomplex__of__bounded__linear,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( real_V1632203528linear @ A @ complex @ ( poinca1531302022lex_of @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_123_c1__on__open__R2_Ocomplex__of__linear,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( real_Vector_linear @ A @ complex @ ( poinca1531302022lex_of @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_124_c1__on__open__R2_Oreal__of__linear,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( real_Vector_linear @ complex @ A @ ( poinca179318102eal_of @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_125_comp__apply__eq,axiom,
% 3.76/2.02 ! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,X2: C,H: D > A,K: C > D] :
% 3.76/2.02 ( ( ( F2 @ ( G2 @ X2 ) )
% 3.76/2.02 = ( H @ ( K @ X2 ) ) )
% 3.76/2.02 => ( ( comp @ B @ A @ C @ F2 @ G2 @ X2 )
% 3.76/2.02 = ( comp @ D @ A @ C @ H @ K @ X2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_126_c1__on__open__R2_Oreal__of__inj,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( inj_on @ complex @ A @ ( poinca179318102eal_of @ A ) @ ( top_top @ ( set @ complex ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_127_c1__on__open__R2_Oreal__of__complex__of,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( ( comp @ complex @ A @ A @ ( poinca179318102eal_of @ A ) @ ( poinca1531302022lex_of @ A ) )
% 3.76/2.02 = ( id @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_128_c1__on__open__R2_Ocomplex__of__bij,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
% 3.76/2.02 ( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
% 3.76/2.02 => ( bij_betw @ A @ complex @ ( poinca1531302022lex_of @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ complex ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_129_the__inv__f__o__f__id,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,Z3: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 @ Z3 )
% 3.76/2.02 = ( id @ A @ Z3 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_130_linear__surjective__right__inverse,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType] :
% 3.76/2.02 ( ( ( real_V1076094709vector @ B )
% 3.76/2.02 & ( real_V1076094709vector @ A ) )
% 3.76/2.02 => ! [F2: A > B] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ? [G4: B > A] :
% 3.76/2.02 ( ( real_Vector_linear @ B @ A @ G4 )
% 3.76/2.02 & ( ( comp @ A @ B @ B @ F2 @ G4 )
% 3.76/2.02 = ( id @ B ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_131_iso__tuple__update__accessor__eq__assist__idI,axiom,
% 3.76/2.02 ! [A: $tType,V3: A,F2: A > A,V: A] :
% 3.76/2.02 ( ( V3
% 3.76/2.02 = ( F2 @ V ) )
% 3.76/2.02 => ( iso_tu2011167877assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) @ V @ F2 @ V3 @ V ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_132_bij__swap__compose__bij,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,P: A > B,A2: B,B2: B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( bij_betw @ A @ B @ ( comp @ B @ B @ A @ ( swap @ B @ B @ A2 @ B2 @ ( id @ B ) ) @ P ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_133_image__eqI,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,B2: A,F2: B > A,X2: B,A4: set @ B] :
% 3.76/2.02 ( ( B2
% 3.76/2.02 = ( F2 @ X2 ) )
% 3.76/2.02 => ( ( member @ B @ X2 @ A4 )
% 3.76/2.02 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_134_swap__nilpotent,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A2: A,B2: A,F2: A > B] :
% 3.76/2.02 ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ A2 @ B2 @ F2 ) )
% 3.76/2.02 = F2 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_135_swap__self,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A2: A,F2: A > B] :
% 3.76/2.02 ( ( swap @ A @ B @ A2 @ A2 @ F2 )
% 3.76/2.02 = F2 ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_136_swap__apply_I3_J,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,C2: B,A2: B,B2: B,F2: B > A] :
% 3.76/2.02 ( ( C2 != A2 )
% 3.76/2.02 => ( ( C2 != B2 )
% 3.76/2.02 => ( ( swap @ B @ A @ A2 @ B2 @ F2 @ C2 )
% 3.76/2.02 = ( F2 @ C2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_137_image__id,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( image @ A @ A @ ( id @ A ) )
% 3.76/2.02 = ( id @ ( set @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_138_swap__image__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: A,F2: A > B] :
% 3.76/2.02 ( ( member @ A @ A2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ B2 @ A4 )
% 3.76/2.02 => ( ( image @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 )
% 3.76/2.02 = ( image @ A @ B @ F2 @ A4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_139_inj__on__swap__iff,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: A,F2: A > B] :
% 3.76/2.02 ( ( member @ A @ A2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ B2 @ A4 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 )
% 3.76/2.02 = ( inj_on @ A @ B @ F2 @ A4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_140_bij__betw__swap__iff,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,X2: A,A4: set @ A,Y: A,F2: A > B,B4: set @ B] :
% 3.76/2.02 ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ Y @ A4 )
% 3.76/2.02 => ( ( bij_betw @ A @ B @ ( swap @ A @ B @ X2 @ Y @ F2 ) @ A4 @ B4 )
% 3.76/2.02 = ( bij_betw @ A @ B @ F2 @ A4 @ B4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_141_swap__comp__involutory,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A2: A,B2: A] :
% 3.76/2.02 ( ( comp @ ( A > B ) @ ( A > B ) @ ( A > B ) @ ( swap @ A @ B @ A2 @ B2 ) @ ( swap @ A @ B @ A2 @ B2 ) )
% 3.76/2.02 = ( id @ ( A > B ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_142_surj__swap__iff,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A2: B,B2: B,F2: B > A] :
% 3.76/2.02 ( ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_143_bij__swap__iff,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,A2: A,B2: A,F2: A > B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_144_swap__id__idempotent,axiom,
% 3.76/2.02 ! [A: $tType,A2: A,B2: A] :
% 3.76/2.02 ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) )
% 3.76/2.02 = ( id @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_145_the__inv__into__onto,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ ( image @ A @ B @ F2 @ A4 ) )
% 3.76/2.02 = A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_146_swap__id__eq,axiom,
% 3.76/2.02 ! [A: $tType,X2: A,A2: A,B2: A] :
% 3.76/2.02 ( ( ( X2 = A2 )
% 3.76/2.02 => ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
% 3.76/2.02 = B2 ) )
% 3.76/2.02 & ( ( X2 != A2 )
% 3.76/2.02 => ( ( ( X2 = B2 )
% 3.76/2.02 => ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
% 3.76/2.02 = A2 ) )
% 3.76/2.02 & ( ( X2 != B2 )
% 3.76/2.02 => ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
% 3.76/2.02 = X2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_147_comp__swap,axiom,
% 3.76/2.02 ! [B: $tType,C: $tType,A: $tType,F2: C > B,A2: A,B2: A,G2: A > C] :
% 3.76/2.02 ( ( comp @ C @ B @ A @ F2 @ ( swap @ A @ C @ A2 @ B2 @ G2 ) )
% 3.76/2.02 = ( swap @ A @ B @ A2 @ B2 @ ( comp @ C @ B @ A @ F2 @ G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_148_the__inv__into__comp,axiom,
% 3.76/2.02 ! [A: $tType,C: $tType,B: $tType,F2: A > B,G2: C > A,A4: set @ C,X2: B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( image @ C @ A @ G2 @ A4 ) )
% 3.76/2.02 => ( ( inj_on @ C @ A @ G2 @ A4 )
% 3.76/2.02 => ( ( member @ B @ X2 @ ( image @ A @ B @ F2 @ ( image @ C @ A @ G2 @ A4 ) ) )
% 3.76/2.02 => ( ( the_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F2 @ G2 ) @ X2 )
% 3.76/2.02 = ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A4 @ G2 ) @ ( the_inv_into @ A @ B @ ( image @ C @ A @ G2 @ A4 ) @ F2 ) @ X2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_149_image__eq__imp__comp,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,D: $tType,C: $tType,F2: B > A,A4: set @ B,G2: C > A,B4: set @ C,H: A > D] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ A4 )
% 3.76/2.02 = ( image @ C @ A @ G2 @ B4 ) )
% 3.76/2.02 => ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F2 ) @ A4 )
% 3.76/2.02 = ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G2 ) @ B4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_150_image__comp,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,R: set @ C] :
% 3.76/2.02 ( ( image @ B @ A @ F2 @ ( image @ C @ B @ G2 @ R ) )
% 3.76/2.02 = ( image @ C @ A @ ( comp @ B @ A @ C @ F2 @ G2 ) @ R ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_151_bij__betw__imp__surj__on,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ( ( image @ A @ B @ F2 @ A4 )
% 3.76/2.02 = B4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_152_surj__imp__surj__swap,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: B > A,A2: B,B2: B] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_153_inj__on__the__inv__into,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_154_f__the__inv__into__f,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,Y: B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( member @ B @ Y @ ( image @ A @ B @ F2 @ A4 ) )
% 3.76/2.02 => ( ( F2 @ ( the_inv_into @ A @ B @ A4 @ F2 @ Y ) )
% 3.76/2.02 = Y ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_155_swap__commute,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( swap @ A @ B )
% 3.76/2.02 = ( ^ [A7: A,B3: A] : ( swap @ A @ B @ B3 @ A7 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_156_swap__triple,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A2: A,C2: A,B2: A,F2: A > B] :
% 3.76/2.02 ( ( A2 != C2 )
% 3.76/2.02 => ( ( B2 != C2 )
% 3.76/2.02 => ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ B2 @ C2 @ ( swap @ A @ B @ A2 @ B2 @ F2 ) ) )
% 3.76/2.02 = ( swap @ A @ B @ A2 @ C2 @ F2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_157_iso__tuple__update__accessor__eq__assist__triv,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
% 3.76/2.02 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
% 3.76/2.02 => ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_158_update__accessor__accessor__eqE,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
% 3.76/2.02 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
% 3.76/2.02 => ( ( Ac @ V )
% 3.76/2.02 = X2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_159_update__accessor__updator__eqE,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
% 3.76/2.02 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
% 3.76/2.02 => ( ( Upd @ F2 @ V )
% 3.76/2.02 = V3 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_160_rev__image__eqI,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,X2: A,A4: set @ A,B2: B,F2: A > B] :
% 3.76/2.02 ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( B2
% 3.76/2.02 = ( F2 @ X2 ) )
% 3.76/2.02 => ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_161_ball__imageD,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P2: A > $o] :
% 3.76/2.02 ( ! [X4: A] :
% 3.76/2.02 ( ( member @ A @ X4 @ ( image @ B @ A @ F2 @ A4 ) )
% 3.76/2.02 => ( P2 @ X4 ) )
% 3.76/2.02 => ! [X5: B] :
% 3.76/2.02 ( ( member @ B @ X5 @ A4 )
% 3.76/2.02 => ( P2 @ ( F2 @ X5 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_162_image__cong,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F2: A > B,G2: A > B] :
% 3.76/2.02 ( ( M = N )
% 3.76/2.02 => ( ! [X4: A] :
% 3.76/2.02 ( ( member @ A @ X4 @ N )
% 3.76/2.02 => ( ( F2 @ X4 )
% 3.76/2.02 = ( G2 @ X4 ) ) )
% 3.76/2.02 => ( ( image @ A @ B @ F2 @ M )
% 3.76/2.02 = ( image @ A @ B @ G2 @ N ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_163_bex__imageD,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P2: A > $o] :
% 3.76/2.02 ( ? [X5: A] :
% 3.76/2.02 ( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A4 ) )
% 3.76/2.02 & ( P2 @ X5 ) )
% 3.76/2.02 => ? [X4: B] :
% 3.76/2.02 ( ( member @ B @ X4 @ A4 )
% 3.76/2.02 & ( P2 @ ( F2 @ X4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_164_image__iff,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,Z3: A,F2: B > A,A4: set @ B] :
% 3.76/2.02 ( ( member @ A @ Z3 @ ( image @ B @ A @ F2 @ A4 ) )
% 3.76/2.02 = ( ? [X: B] :
% 3.76/2.02 ( ( member @ B @ X @ A4 )
% 3.76/2.02 & ( Z3
% 3.76/2.02 = ( F2 @ X ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_165_imageI,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,X2: A,A4: set @ A,F2: A > B] :
% 3.76/2.02 ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( member @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_166_inj__on__image__iff,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,A4: set @ A,G2: A > B,F2: A > A] :
% 3.76/2.02 ( ! [X4: A] :
% 3.76/2.02 ( ( member @ A @ X4 @ A4 )
% 3.76/2.02 => ! [Xa: A] :
% 3.76/2.02 ( ( member @ A @ Xa @ A4 )
% 3.76/2.02 => ( ( ( G2 @ ( F2 @ X4 ) )
% 3.76/2.02 = ( G2 @ ( F2 @ Xa ) ) )
% 3.76/2.02 = ( ( G2 @ X4 )
% 3.76/2.02 = ( G2 @ Xa ) ) ) ) )
% 3.76/2.02 => ( ( inj_on @ A @ A @ F2 @ A4 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ G2 @ ( image @ A @ A @ F2 @ A4 ) )
% 3.76/2.02 = ( inj_on @ A @ B @ G2 @ A4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_167_surjD,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: B > A,Y: A] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ? [X4: B] :
% 3.76/2.02 ( Y
% 3.76/2.02 = ( F2 @ X4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_168_surjE,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: B > A,Y: A] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ~ ! [X4: B] :
% 3.76/2.02 ( Y
% 3.76/2.02 != ( F2 @ X4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_169_surjI,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,G2: B > A,F2: A > B] :
% 3.76/2.02 ( ! [X4: A] :
% 3.76/2.02 ( ( G2 @ ( F2 @ X4 ) )
% 3.76/2.02 = X4 )
% 3.76/2.02 => ( ( image @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_170_surj__def,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: B > A] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( ! [Y3: A] :
% 3.76/2.02 ? [X: B] :
% 3.76/2.02 ( Y3
% 3.76/2.02 = ( F2 @ X ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_171_rangeI,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: B > A,X2: B] : ( member @ A @ ( F2 @ X2 ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_172_range__eqI,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,B2: A,F2: B > A,X2: B] :
% 3.76/2.02 ( ( B2
% 3.76/2.02 = ( F2 @ X2 ) )
% 3.76/2.02 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_173_inj__on__imp__inj__on__swap,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A2: A,B2: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ A2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ B2 @ A4 )
% 3.76/2.02 => ( inj_on @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_174_the__inv__into__f__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( ( F2 @ X2 )
% 3.76/2.02 = Y )
% 3.76/2.02 => ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( the_inv_into @ A @ B @ A4 @ F2 @ Y )
% 3.76/2.02 = X2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_175_the__inv__into__f__f,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( the_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ X2 ) )
% 3.76/2.02 = X2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_176_f__the__inv__into__f__bij__betw,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B,X2: B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ( ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ( member @ B @ X2 @ B4 ) )
% 3.76/2.02 => ( ( F2 @ ( the_inv_into @ A @ B @ A4 @ F2 @ X2 ) )
% 3.76/2.02 = X2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_177_bij__betw__the__inv__into,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ( bij_betw @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ B4 @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_178_swap__general,axiom,
% 3.76/2.02 ! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
% 3.76/2.02 ( ( A2 != B2 )
% 3.76/2.02 => ( ( C2 != D2 )
% 3.76/2.02 => ( ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
% 3.76/2.02 = ( id @ A ) )
% 3.76/2.02 | ? [X4: A,Y2: A,Z4: A] :
% 3.76/2.02 ( ( X4 != A2 )
% 3.76/2.02 & ( Y2 != A2 )
% 3.76/2.02 & ( Z4 != A2 )
% 3.76/2.02 & ( X4 != Y2 )
% 3.76/2.02 & ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
% 3.76/2.02 = ( comp @ A @ A @ A @ ( swap @ A @ A @ X4 @ Y2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ Z4 @ ( id @ A ) ) ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_179_swap__id__common,axiom,
% 3.76/2.02 ! [A: $tType,A2: A,C2: A,B2: A] :
% 3.76/2.02 ( ( A2 != C2 )
% 3.76/2.02 => ( ( B2 != C2 )
% 3.76/2.02 => ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ C2 @ ( id @ A ) ) )
% 3.76/2.02 = ( comp @ A @ A @ A @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_180_swap__id__common_H,axiom,
% 3.76/2.02 ! [A: $tType,A2: A,B2: A,C2: A] :
% 3.76/2.02 ( ( A2 != B2 )
% 3.76/2.02 => ( ( A2 != C2 )
% 3.76/2.02 => ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) )
% 3.76/2.02 = ( comp @ A @ A @ A @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_181_swap__id__independent,axiom,
% 3.76/2.02 ! [A: $tType,A2: A,C2: A,D2: A,B2: A] :
% 3.76/2.02 ( ( A2 != C2 )
% 3.76/2.02 => ( ( A2 != D2 )
% 3.76/2.02 => ( ( B2 != C2 )
% 3.76/2.02 => ( ( B2 != D2 )
% 3.76/2.02 => ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
% 3.76/2.02 = ( comp @ A @ A @ A @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_182_fun_Oinj__map__strong,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,D: $tType,X2: D > A,Xa2: D > A,F2: A > B,Fa: A > B] :
% 3.76/2.02 ( ! [Z4: A,Za: A] :
% 3.76/2.02 ( ( member @ A @ Z4 @ ( image @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
% 3.76/2.02 => ( ( member @ A @ Za @ ( image @ D @ A @ Xa2 @ ( top_top @ ( set @ D ) ) ) )
% 3.76/2.02 => ( ( ( F2 @ Z4 )
% 3.76/2.02 = ( Fa @ Za ) )
% 3.76/2.02 => ( Z4 = Za ) ) ) )
% 3.76/2.02 => ( ( ( comp @ A @ B @ D @ F2 @ X2 )
% 3.76/2.02 = ( comp @ A @ B @ D @ Fa @ Xa2 ) )
% 3.76/2.02 => ( X2 = Xa2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_183_fun_Omap__cong0,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,D: $tType,X2: D > A,F2: A > B,G2: A > B] :
% 3.76/2.02 ( ! [Z4: A] :
% 3.76/2.02 ( ( member @ A @ Z4 @ ( image @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
% 3.76/2.02 => ( ( F2 @ Z4 )
% 3.76/2.02 = ( G2 @ Z4 ) ) )
% 3.76/2.02 => ( ( comp @ A @ B @ D @ F2 @ X2 )
% 3.76/2.02 = ( comp @ A @ B @ D @ G2 @ X2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_184_fun_Omap__cong,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,D: $tType,X2: D > A,Ya: D > A,F2: A > B,G2: A > B] :
% 3.76/2.02 ( ( X2 = Ya )
% 3.76/2.02 => ( ! [Z4: A] :
% 3.76/2.02 ( ( member @ A @ Z4 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 3.76/2.02 => ( ( F2 @ Z4 )
% 3.76/2.02 = ( G2 @ Z4 ) ) )
% 3.76/2.02 => ( ( comp @ A @ B @ D @ F2 @ X2 )
% 3.76/2.02 = ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_185_fun_Oset__map,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,D: $tType,F2: A > B,V: D > A] :
% 3.76/2.02 ( ( image @ D @ B @ ( comp @ A @ B @ D @ F2 @ V ) @ ( top_top @ ( set @ D ) ) )
% 3.76/2.02 = ( image @ A @ B @ F2 @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_186_comp__surj,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: A > C] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( ( image @ A @ C @ G2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ C ) ) )
% 3.76/2.02 => ( ( image @ B @ C @ ( comp @ A @ C @ B @ G2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_187_inj__image__mem__iff,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A2: A,A4: set @ A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ A4 ) )
% 3.76/2.02 = ( member @ A @ A2 @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_188_inj__image__eq__iff,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( ( image @ A @ B @ F2 @ A4 )
% 3.76/2.02 = ( image @ A @ B @ F2 @ B4 ) )
% 3.76/2.02 = ( A4 = B4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_189_range__ex1__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,B2: B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( member @ B @ B2 @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
% 3.76/2.02 = ( ? [X: A] :
% 3.76/2.02 ( ( B2
% 3.76/2.02 = ( F2 @ X ) )
% 3.76/2.02 & ! [Y3: A] :
% 3.76/2.02 ( ( B2
% 3.76/2.02 = ( F2 @ Y3 ) )
% 3.76/2.02 => ( Y3 = X ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_190_surj__id,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( image @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_191_comp__inj__on__iff,axiom,
% 3.76/2.02 ! [B: $tType,C: $tType,A: $tType,F2: A > B,A4: set @ A,F3: B > C] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( inj_on @ B @ C @ F3 @ ( image @ A @ B @ F2 @ A4 ) )
% 3.76/2.02 = ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F3 @ F2 ) @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_192_inj__on__imageI,axiom,
% 3.76/2.02 ! [B: $tType,C: $tType,A: $tType,G2: C > B,F2: A > C,A4: set @ A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G2 @ F2 ) @ A4 )
% 3.76/2.02 => ( inj_on @ C @ B @ G2 @ ( image @ A @ C @ F2 @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_193_comp__inj__on,axiom,
% 3.76/2.02 ! [B: $tType,C: $tType,A: $tType,F2: A > B,A4: set @ A,G2: B > C] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( inj_on @ B @ C @ G2 @ ( image @ A @ B @ F2 @ A4 ) )
% 3.76/2.02 => ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_194_bij__betw__imp__surj,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ B ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_195_bij__is__surj,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ B ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_196_inj__on__imp__bij__betw,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( bij_betw @ A @ B @ F2 @ A4 @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_197_bij__betw__imageI,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( ( image @ A @ B @ F2 @ A4 )
% 3.76/2.02 = B4 )
% 3.76/2.02 => ( bij_betw @ A @ B @ F2 @ A4 @ B4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_198_bij__betw__def,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( bij_betw @ A @ B )
% 3.76/2.02 = ( ^ [F: A > B,A5: set @ A,B5: set @ B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F @ A5 )
% 3.76/2.02 & ( ( image @ A @ B @ F @ A5 )
% 3.76/2.02 = B5 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_199_the__inv__f__f,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,X2: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X2 ) )
% 3.76/2.02 = X2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_200_bij__def,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 & ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_201_bijI,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_202_linear__surj__imp__inj,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( euclid925273238_space @ A )
% 3.76/2.02 => ! [F2: A > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ A @ F2 )
% 3.76/2.02 => ( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_203_linear__inj__imp__surj,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( euclid925273238_space @ A )
% 3.76/2.02 => ! [F2: A > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ A @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_204_surj__fun__eq,axiom,
% 3.76/2.02 ! [B: $tType,C: $tType,A: $tType,F2: B > A,X7: set @ B,G1: A > C,G22: A > C] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ X7 )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ! [X4: B] :
% 3.76/2.02 ( ( member @ B @ X4 @ X7 )
% 3.76/2.02 => ( ( comp @ A @ C @ B @ G1 @ F2 @ X4 )
% 3.76/2.02 = ( comp @ A @ C @ B @ G22 @ F2 @ X4 ) ) )
% 3.76/2.02 => ( G1 = G22 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_205_Inf_OINF__id__eq,axiom,
% 3.76/2.02 ! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
% 3.76/2.02 ( ( Inf @ ( image @ A @ A @ ( id @ A ) @ A4 ) )
% 3.76/2.02 = ( Inf @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_206_Sup_OSUP__id__eq,axiom,
% 3.76/2.02 ! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
% 3.76/2.02 ( ( Sup @ ( image @ A @ A @ ( id @ A ) @ A4 ) )
% 3.76/2.02 = ( Sup @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_207_Inf_OINF__image,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G2: B > A,F2: C > B,A4: set @ C] :
% 3.76/2.02 ( ( Inf @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F2 @ A4 ) ) )
% 3.76/2.02 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_208_bij__betwI_H,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,X7: set @ A,F2: A > B,Y5: set @ B] :
% 3.76/2.02 ( ! [X4: A] :
% 3.76/2.02 ( ( member @ A @ X4 @ X7 )
% 3.76/2.02 => ! [Y2: A] :
% 3.76/2.02 ( ( member @ A @ Y2 @ X7 )
% 3.76/2.02 => ( ( ( F2 @ X4 )
% 3.76/2.02 = ( F2 @ Y2 ) )
% 3.76/2.02 = ( X4 = Y2 ) ) ) )
% 3.76/2.02 => ( ! [X4: A] :
% 3.76/2.02 ( ( member @ A @ X4 @ X7 )
% 3.76/2.02 => ( member @ B @ ( F2 @ X4 ) @ Y5 ) )
% 3.76/2.02 => ( ! [Y2: B] :
% 3.76/2.02 ( ( member @ B @ Y2 @ Y5 )
% 3.76/2.02 => ? [X5: A] :
% 3.76/2.02 ( ( member @ A @ X5 @ X7 )
% 3.76/2.02 & ( Y2
% 3.76/2.02 = ( F2 @ X5 ) ) ) )
% 3.76/2.02 => ( bij_betw @ A @ B @ F2 @ X7 @ Y5 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_209_Sup_OSUP__image,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G2: B > A,F2: C > B,A4: set @ C] :
% 3.76/2.02 ( ( Sup @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F2 @ A4 ) ) )
% 3.76/2.02 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ A4 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_210_inj__adjoint__iff__surj,axiom,
% 3.76/2.02 ! [M2: $tType,N2: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ N2 )
% 3.76/2.02 & ( euclid925273238_space @ M2 ) )
% 3.76/2.02 => ! [F2: M2 > N2] :
% 3.76/2.02 ( ( real_Vector_linear @ M2 @ N2 @ F2 )
% 3.76/2.02 => ( ( inj_on @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
% 3.76/2.02 = ( ( image @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) )
% 3.76/2.02 = ( top_top @ ( set @ N2 ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_211_surj__adjoint__iff__inj,axiom,
% 3.76/2.02 ! [N2: $tType,M2: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ M2 )
% 3.76/2.02 & ( euclid925273238_space @ N2 ) )
% 3.76/2.02 => ! [F2: M2 > N2] :
% 3.76/2.02 ( ( real_Vector_linear @ M2 @ N2 @ F2 )
% 3.76/2.02 => ( ( ( image @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
% 3.76/2.02 = ( top_top @ ( set @ M2 ) ) )
% 3.76/2.02 = ( inj_on @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_212_adjoint__adjoint,axiom,
% 3.76/2.02 ! [M2: $tType,N2: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ N2 )
% 3.76/2.02 & ( euclid925273238_space @ M2 ) )
% 3.76/2.02 => ! [F2: N2 > M2] :
% 3.76/2.02 ( ( real_Vector_linear @ N2 @ M2 @ F2 )
% 3.76/2.02 => ( ( linear_adjoint @ M2 @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 ) )
% 3.76/2.02 = F2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_213_adjoint__linear,axiom,
% 3.76/2.02 ! [M2: $tType,N2: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ N2 )
% 3.76/2.02 & ( euclid925273238_space @ M2 ) )
% 3.76/2.02 => ! [F2: N2 > M2] :
% 3.76/2.02 ( ( real_Vector_linear @ N2 @ M2 @ F2 )
% 3.76/2.02 => ( real_Vector_linear @ M2 @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_214_linear__surj__adj__imp__inj,axiom,
% 3.76/2.02 ! [N2: $tType,M2: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ M2 )
% 3.76/2.02 & ( euclid925273238_space @ N2 ) )
% 3.76/2.02 => ! [F2: M2 > N2] :
% 3.76/2.02 ( ( real_Vector_linear @ M2 @ N2 @ F2 )
% 3.76/2.02 => ( ( ( image @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
% 3.76/2.02 = ( top_top @ ( set @ M2 ) ) )
% 3.76/2.02 => ( inj_on @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_215_coplanar__linear__image__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ A )
% 3.76/2.02 & ( euclid925273238_space @ B ) )
% 3.76/2.02 => ! [F2: A > B,S: set @ A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( coplanar @ B @ ( image @ A @ B @ F2 @ S ) )
% 3.76/2.02 = ( coplanar @ A @ S ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_216_linear__singular__into__hyperplane,axiom,
% 3.76/2.02 ! [N2: $tType] :
% 3.76/2.02 ( ( euclid925273238_space @ N2 )
% 3.76/2.02 => ! [F2: N2 > N2] :
% 3.76/2.02 ( ( real_Vector_linear @ N2 @ N2 @ F2 )
% 3.76/2.02 => ( ( ~ ( inj_on @ N2 @ N2 @ F2 @ ( top_top @ ( set @ N2 ) ) ) )
% 3.76/2.02 = ( ? [A7: N2] :
% 3.76/2.02 ( ( A7
% 3.76/2.02 != ( zero_zero @ N2 ) )
% 3.76/2.02 & ! [X: N2] :
% 3.76/2.02 ( ( inner_780170721_inner @ N2 @ A7 @ ( F2 @ X ) )
% 3.76/2.02 = ( zero_zero @ real ) ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_217_adjoint__unique,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType] :
% 3.76/2.02 ( ( ( inner_real_inner @ B )
% 3.76/2.02 & ( inner_real_inner @ A ) )
% 3.76/2.02 => ! [F2: A > B,G2: B > A] :
% 3.76/2.02 ( ! [X4: A,Y2: B] :
% 3.76/2.02 ( ( inner_780170721_inner @ B @ ( F2 @ X4 ) @ Y2 )
% 3.76/2.02 = ( inner_780170721_inner @ A @ X4 @ ( G2 @ Y2 ) ) )
% 3.76/2.02 => ( ( linear_adjoint @ A @ B @ F2 )
% 3.76/2.02 = G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_218_hyperplane__eq__Ex,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( inner_real_inner @ A )
% 3.76/2.02 => ! [A2: A,B2: real] :
% 3.76/2.02 ( ( A2
% 3.76/2.02 != ( zero_zero @ A ) )
% 3.76/2.02 => ~ ! [X4: A] :
% 3.76/2.02 ( ( inner_780170721_inner @ A @ A2 @ X4 )
% 3.76/2.02 != B2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_219_adjoint__works,axiom,
% 3.76/2.02 ! [N2: $tType,M2: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ M2 )
% 3.76/2.02 & ( euclid925273238_space @ N2 ) )
% 3.76/2.02 => ! [F2: N2 > M2,X2: N2,Y: M2] :
% 3.76/2.02 ( ( real_Vector_linear @ N2 @ M2 @ F2 )
% 3.76/2.02 => ( ( inner_780170721_inner @ N2 @ X2 @ ( linear_adjoint @ N2 @ M2 @ F2 @ Y ) )
% 3.76/2.02 = ( inner_780170721_inner @ M2 @ ( F2 @ X2 ) @ Y ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_220_adjoint__clauses_I2_J,axiom,
% 3.76/2.02 ! [M2: $tType,N2: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ N2 )
% 3.76/2.02 & ( euclid925273238_space @ M2 ) )
% 3.76/2.02 => ! [F2: N2 > M2,Y: M2,X2: N2] :
% 3.76/2.02 ( ( real_Vector_linear @ N2 @ M2 @ F2 )
% 3.76/2.02 => ( ( inner_780170721_inner @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 @ Y ) @ X2 )
% 3.76/2.02 = ( inner_780170721_inner @ M2 @ Y @ ( F2 @ X2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_221_vector__eq,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( inner_real_inner @ A )
% 3.76/2.02 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 3.76/2.02 = ( ^ [X: A,Y3: A] :
% 3.76/2.02 ( ( ( inner_780170721_inner @ A @ X @ X )
% 3.76/2.02 = ( inner_780170721_inner @ A @ X @ Y3 ) )
% 3.76/2.02 & ( ( inner_780170721_inner @ A @ Y3 @ Y3 )
% 3.76/2.02 = ( inner_780170721_inner @ A @ X @ X ) ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_222_vector__eq__ldot,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( inner_real_inner @ A )
% 3.76/2.02 => ! [Y: A,Z3: A] :
% 3.76/2.02 ( ( ! [X: A] :
% 3.76/2.02 ( ( inner_780170721_inner @ A @ X @ Y )
% 3.76/2.02 = ( inner_780170721_inner @ A @ X @ Z3 ) ) )
% 3.76/2.02 = ( Y = Z3 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_223_vector__eq__rdot,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( inner_real_inner @ A )
% 3.76/2.02 => ! [X2: A,Y: A] :
% 3.76/2.02 ( ( ! [Z2: A] :
% 3.76/2.02 ( ( inner_780170721_inner @ A @ X2 @ Z2 )
% 3.76/2.02 = ( inner_780170721_inner @ A @ Y @ Z2 ) ) )
% 3.76/2.02 = ( X2 = Y ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_224_coplanar__linear__image,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ A )
% 3.76/2.02 & ( real_V55928688vector @ B ) )
% 3.76/2.02 => ! [S: set @ A,F2: A > B] :
% 3.76/2.02 ( ( coplanar @ A @ S )
% 3.76/2.02 => ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( coplanar @ B @ ( image @ A @ B @ F2 @ S ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_225_inner__eq__zero__iff,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( inner_real_inner @ A )
% 3.76/2.02 => ! [X2: A] :
% 3.76/2.02 ( ( ( inner_780170721_inner @ A @ X2 @ X2 )
% 3.76/2.02 = ( zero_zero @ real ) )
% 3.76/2.02 = ( X2
% 3.76/2.02 = ( zero_zero @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_226_inner__zero__right,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( inner_real_inner @ A )
% 3.76/2.02 => ! [X2: A] :
% 3.76/2.02 ( ( inner_780170721_inner @ A @ X2 @ ( zero_zero @ A ) )
% 3.76/2.02 = ( zero_zero @ real ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_227_all__zero__iff,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( inner_real_inner @ A )
% 3.76/2.02 => ! [X2: A] :
% 3.76/2.02 ( ( ! [U: A] :
% 3.76/2.02 ( ( inner_780170721_inner @ A @ X2 @ U )
% 3.76/2.02 = ( zero_zero @ real ) ) )
% 3.76/2.02 = ( X2
% 3.76/2.02 = ( zero_zero @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_228_inner__zero__left,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( inner_real_inner @ A )
% 3.76/2.02 => ! [X2: A] :
% 3.76/2.02 ( ( inner_780170721_inner @ A @ ( zero_zero @ A ) @ X2 )
% 3.76/2.02 = ( zero_zero @ real ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_229_aff__dim__injective__linear__image,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType] :
% 3.76/2.02 ( ( ( euclid925273238_space @ A )
% 3.76/2.02 & ( euclid925273238_space @ B ) )
% 3.76/2.02 => ! [F2: A > B,S: set @ A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ B @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( aff_dim @ B @ ( image @ A @ B @ F2 @ S ) )
% 3.76/2.02 = ( aff_dim @ A @ S ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_230_bij__swap__comp,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,P: A > B,A2: B,B2: B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( ( comp @ B @ B @ A @ ( swap @ B @ B @ A2 @ B2 @ ( id @ B ) ) @ P )
% 3.76/2.02 = ( swap @ A @ B @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ A2 ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ B2 ) @ P ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_231_inj__linear__imp__inv__bounded__linear,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( euclid925273238_space @ A )
% 3.76/2.02 => ! [F2: A > A] :
% 3.76/2.02 ( ( real_V1632203528linear @ A @ A @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( real_V1632203528linear @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_232_inv__swap__id,axiom,
% 3.76/2.02 ! [A: $tType,A2: A,B2: A] :
% 3.76/2.02 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) )
% 3.76/2.02 = ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_233_inj__linear__imp__inv__linear,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( euclid925273238_space @ A )
% 3.76/2.02 => ! [F2: A > A] :
% 3.76/2.02 ( ( real_Vector_linear @ A @ A @ F2 )
% 3.76/2.02 => ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( real_Vector_linear @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_234_inv__o__cancel,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
% 3.76/2.02 = ( id @ A ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_235_o__inv__o__cancel,axiom,
% 3.76/2.02 ! [B: $tType,C: $tType,A: $tType,F2: A > B,G2: A > C] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G2 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ F2 )
% 3.76/2.02 = G2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_236_inv__into__f__f,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ X2 ) )
% 3.76/2.02 = X2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_237_inv__id,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
% 3.76/2.02 = ( id @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_238_inv__equality,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,G2: B > A,F2: A > B] :
% 3.76/2.02 ( ! [X4: A] :
% 3.76/2.02 ( ( G2 @ ( F2 @ X4 ) )
% 3.76/2.02 = X4 )
% 3.76/2.02 => ( ! [Y2: B] :
% 3.76/2.02 ( ( F2 @ ( G2 @ Y2 ) )
% 3.76/2.02 = Y2 )
% 3.76/2.02 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
% 3.76/2.02 = G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_239_inv__into__f__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.76/2.02 => ( ( member @ A @ X2 @ A4 )
% 3.76/2.02 => ( ( ( F2 @ X2 )
% 3.76/2.02 = Y )
% 3.76/2.02 => ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ Y )
% 3.76/2.02 = X2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_240_bij__betw__inv__into,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
% 3.76/2.02 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ B4 @ A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_241_inv__into__inv__into__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A8: set @ B,A2: A] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 3.76/2.02 => ( ( member @ A @ A2 @ A4 )
% 3.76/2.02 => ( ( hilbert_inv_into @ B @ A @ A8 @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ A2 )
% 3.76/2.02 = ( F2 @ A2 ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_242_bij__betw__inv__into__left,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A8: set @ B,A2: A] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 3.76/2.02 => ( ( member @ A @ A2 @ A4 )
% 3.76/2.02 => ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ A2 ) )
% 3.76/2.02 = A2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_243_bij__betw__inv__into__right,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,A4: set @ A,A8: set @ B,A10: B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
% 3.76/2.02 => ( ( member @ B @ A10 @ A8 )
% 3.76/2.02 => ( ( F2 @ ( hilbert_inv_into @ A @ B @ A4 @ F2 @ A10 ) )
% 3.76/2.02 = A10 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_244_surj__f__inv__f,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: B > A,Y: A] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ Y ) )
% 3.76/2.02 = Y ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_245_surj__iff__all,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: B > A] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 = ( ! [X: A] :
% 3.76/2.02 ( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ X ) )
% 3.76/2.02 = X ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_246_image__f__inv__f,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: B > A,A4: set @ A] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( image @ B @ A @ F2 @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ A4 ) )
% 3.76/2.02 = A4 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_247_surj__imp__inv__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: B > A,G2: A > B] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ! [X4: B] :
% 3.76/2.02 ( ( G2 @ ( F2 @ X4 ) )
% 3.76/2.02 = X4 )
% 3.76/2.02 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
% 3.76/2.02 = G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_248_inv__f__f,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,X2: A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X2 ) )
% 3.76/2.02 = X2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_249_inv__f__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B,X2: A,Y: B] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ( ( F2 @ X2 )
% 3.76/2.02 = Y )
% 3.76/2.02 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y )
% 3.76/2.02 = X2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_250_inj__imp__inv__eq,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,F2: A > B,G2: B > A] :
% 3.76/2.02 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( ! [X4: B] :
% 3.76/2.02 ( ( F2 @ ( G2 @ X4 ) )
% 3.76/2.02 = X4 )
% 3.76/2.02 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
% 3.76/2.02 = G2 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_251_bij__imp__bij__inv,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_252_bij__inv__eq__iff,axiom,
% 3.76/2.02 ! [A: $tType,B: $tType,P: A > B,X2: A,Y: B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( ( X2
% 3.76/2.02 = ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ Y ) )
% 3.76/2.02 = ( ( P @ X2 )
% 3.76/2.02 = Y ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_253_inv__inv__eq,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: A > B] :
% 3.76/2.02 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) )
% 3.76/2.02 = F2 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(fact_254_surj__imp__inj__inv,axiom,
% 3.76/2.02 ! [B: $tType,A: $tType,F2: B > A] :
% 3.76/2.02 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
% 3.76/2.02 = ( top_top @ ( set @ A ) ) )
% 3.76/2.02 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___HOL_Otype,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ( type @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ( zero @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Inner__Product_Oreal__inner,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ( inner_real_inner @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Oreal__vector,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ( real_V1076094709vector @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Euclidean__Space_Oeuclidean__space,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ( euclid925273238_space @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Topological__Spaces_Otopological__space,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ( topolo503727757_space @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 3.76/2.02 ! [A: $tType] :
% 3.76/2.02 ( ( execut510477386_space @ A )
% 3.76/2.02 => ( real_V55928688vector @ A ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 3.76/2.02 ! [A11: $tType,A12: $tType] :
% 3.76/2.02 ( ( topolo503727757_space @ A12 )
% 3.76/2.02 => ( topolo503727757_space @ ( A11 > A12 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_fun___Orderings_Otop,axiom,
% 3.76/2.02 ! [A11: $tType,A12: $tType] :
% 3.76/2.02 ( ( top @ A12 )
% 3.76/2.02 => ( top @ ( A11 > A12 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_1,axiom,
% 3.76/2.02 topolo503727757_space @ int ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 3.76/2.02 zero @ int ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
% 3.76/2.02 ! [A11: $tType] : ( top @ ( set @ A11 ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Set_Oset___Groups_Ozero_3,axiom,
% 3.76/2.02 ! [A11: $tType] :
% 3.76/2.02 ( ( zero @ A11 )
% 3.76/2.02 => ( zero @ ( set @ A11 ) ) ) ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_4,axiom,
% 3.76/2.02 topolo503727757_space @ $o ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_HOL_Obool___Orderings_Otop_5,axiom,
% 3.76/2.02 top @ $o ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
% 3.76/2.02 execut510477386_space @ real ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 3.76/2.02 real_V55928688vector @ real ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_6,axiom,
% 3.76/2.02 topolo503727757_space @ real ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
% 3.76/2.02 euclid925273238_space @ real ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
% 3.76/2.02 real_V1076094709vector @ real ).
% 3.76/2.02
% 3.76/2.02 thf(tcon_Real_Oreal___Inner__Product_Oreal__inner,axiom,
% 3.76/2.03 inner_real_inner @ real ).
% 3.76/2.03
% 3.76/2.03 thf(tcon_Real_Oreal___Groups_Ozero_7,axiom,
% 3.76/2.03 zero @ real ).
% 3.76/2.03
% 3.76/2.03 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_8,axiom,
% 3.76/2.03 real_V55928688vector @ complex ).
% 3.76/2.03
% 3.76/2.03 thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_9,axiom,
% 3.76/2.03 topolo503727757_space @ complex ).
% 3.76/2.03
% 3.76/2.03 thf(tcon_Complex_Ocomplex___Euclidean__Space_Oeuclidean__space_10,axiom,
% 3.76/2.03 euclid925273238_space @ complex ).
% 3.76/2.03
% 3.76/2.03 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_11,axiom,
% 3.76/2.03 real_V1076094709vector @ complex ).
% 3.76/2.03
% 3.76/2.03 thf(tcon_Complex_Ocomplex___Inner__Product_Oreal__inner_12,axiom,
% 3.76/2.03 inner_real_inner @ complex ).
% 3.76/2.03
% 3.76/2.03 thf(tcon_Complex_Ocomplex___Groups_Ozero_13,axiom,
% 3.76/2.03 zero @ complex ).
% 3.76/2.03
% 3.76/2.03 thf(tfree_0,hypothesis,
% 3.76/2.03 execut510477386_space @ a ).
% 3.76/2.03
% 3.76/2.03 thf(conj_0,conjecture,
% 3.76/2.03 ( ( path_pathfinish @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) )
% 3.76/2.03 = ( path_pathstart @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) ) ) ).
% 3.76/2.03 ------------------------
% 3.93/2.07 % (11758)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/183Mi)
% 3.93/2.07 % (11761)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/2Mi)
% 3.93/2.07 % (11759)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/4Mi)
% 3.93/2.07 % (11763)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/275Mi)
% 3.93/2.07 % (11760)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/27Mi)
% 3.93/2.07 % (11764)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/18Mi)
% 3.93/2.07 % (11762)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/2Mi)
% 3.93/2.07 % (11761)Instruction limit reached!
% 3.93/2.07 % (11761)------------------------------
% 3.93/2.07 % (11761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.07 % (11761)Termination reason: Unknown
% 3.93/2.07 % (11762)Instruction limit reached!
% 3.93/2.07 % (11762)------------------------------
% 3.93/2.07 % (11762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.07 % (11761)Termination phase: shuffling
% 3.93/2.07
% 3.93/2.07 % (11761)Memory used [KB]: 1407
% 3.93/2.07 % (11761)Time elapsed: 0.003 s
% 3.93/2.07 % (11761)Instructions burned: 2 (million)
% 3.93/2.07 % (11761)------------------------------
% 3.93/2.07 % (11761)------------------------------
% 3.93/2.07 % (11762)Termination reason: Unknown
% 3.93/2.07 % (11762)Termination phase: shuffling
% 3.93/2.07
% 3.93/2.07 % (11762)Memory used [KB]: 1407
% 3.93/2.07 % (11762)Time elapsed: 0.004 s
% 3.93/2.07 % (11762)Instructions burned: 3 (million)
% 3.93/2.07 % (11762)------------------------------
% 3.93/2.07 % (11762)------------------------------
% 3.93/2.07 % (11759)Instruction limit reached!
% 3.93/2.07 % (11759)------------------------------
% 3.93/2.07 % (11759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.07 % (11759)Termination reason: Unknown
% 3.93/2.07 % (11759)Termination phase: shuffling
% 3.93/2.07
% 3.93/2.07 % (11759)Memory used [KB]: 1407
% 3.93/2.07 % (11759)Time elapsed: 0.004 s
% 3.93/2.07 % (11759)Instructions burned: 4 (million)
% 3.93/2.07 % (11759)------------------------------
% 3.93/2.07 % (11759)------------------------------
% 3.93/2.08 % (11764)Instruction limit reached!
% 3.93/2.08 % (11764)------------------------------
% 3.93/2.08 % (11764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.08 % (11764)Termination reason: Unknown
% 3.93/2.08 % (11764)Termination phase: shuffling
% 3.93/2.08
% 3.93/2.08 % (11764)Memory used [KB]: 1791
% 3.93/2.08 % (11764)Time elapsed: 0.013 s
% 3.93/2.08 % (11764)Instructions burned: 19 (million)
% 3.93/2.08 % (11764)------------------------------
% 3.93/2.08 % (11764)------------------------------
% 3.93/2.08 % (11760)Instruction limit reached!
% 3.93/2.08 % (11760)------------------------------
% 3.93/2.08 % (11760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.08 % (11760)Termination reason: Unknown
% 3.93/2.08 % (11760)Termination phase: Property scanning
% 3.93/2.08
% 3.93/2.08 % (11760)Memory used [KB]: 1918
% 3.93/2.08 % (11760)Time elapsed: 0.017 s
% 3.93/2.08 % (11760)Instructions burned: 27 (million)
% 3.93/2.08 % (11760)------------------------------
% 3.93/2.08 % (11760)------------------------------
% 3.93/2.09 % (11765)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/3Mi)
% 3.93/2.09 % (11766)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/37Mi)
% 3.93/2.09 % (11767)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/15Mi)
% 3.93/2.09 % (11765)Instruction limit reached!
% 3.93/2.09 % (11765)------------------------------
% 3.93/2.09 % (11765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.09 % (11765)Termination reason: Unknown
% 3.93/2.09 % (11765)Termination phase: shuffling
% 3.93/2.09
% 3.93/2.09 % (11765)Memory used [KB]: 1407
% 3.93/2.09 % (11765)Time elapsed: 0.004 s
% 3.93/2.09 % (11765)Instructions burned: 3 (million)
% 3.93/2.09 % (11765)------------------------------
% 3.93/2.09 % (11765)------------------------------
% 3.93/2.10 % (11768)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/3Mi)
% 3.93/2.10 % (11767)Instruction limit reached!
% 3.93/2.10 % (11767)------------------------------
% 3.93/2.10 % (11767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.10 % (11767)Termination reason: Unknown
% 3.93/2.10 % (11767)Termination phase: shuffling
% 3.93/2.10
% 3.93/2.10 % (11767)Memory used [KB]: 1663
% 3.93/2.10 % (11767)Time elapsed: 0.011 s
% 3.93/2.10 % (11767)Instructions burned: 16 (million)
% 3.93/2.10 % (11767)------------------------------
% 3.93/2.10 % (11767)------------------------------
% 3.93/2.10 % (11768)Instruction limit reached!
% 3.93/2.10 % (11768)------------------------------
% 3.93/2.10 % (11768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.10 % (11768)Termination reason: Unknown
% 3.93/2.10 % (11768)Termination phase: shuffling
% 3.93/2.10
% 3.93/2.10 % (11768)Memory used [KB]: 1407
% 3.93/2.10 % (11768)Time elapsed: 0.004 s
% 3.93/2.10 % (11768)Instructions burned: 3 (million)
% 3.93/2.10 % (11768)------------------------------
% 3.93/2.10 % (11768)------------------------------
% 3.93/2.10 % (11769)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/1041Mi)
% 3.93/2.11 % (11763)First to succeed.
% 3.93/2.11 % (11766)Instruction limit reached!
% 3.93/2.11 % (11766)------------------------------
% 3.93/2.11 % (11766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.11 % (11766)Termination reason: Unknown
% 3.93/2.11 % (11766)Termination phase: shuffling
% 3.93/2.11
% 3.93/2.11 % (11766)Memory used [KB]: 2174
% 3.93/2.11 % (11766)Time elapsed: 0.022 s
% 3.93/2.11 % (11766)Instructions burned: 38 (million)
% 3.93/2.11 % (11766)------------------------------
% 3.93/2.11 % (11766)------------------------------
% 3.93/2.11 % (11770)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11658 for (2999ds/7Mi)
% 3.93/2.11 % (11763)Refutation found. Thanks to Tanya!
% 3.93/2.11 % SZS status Theorem for DTF2THF_11658
% 3.93/2.11 % SZS output start Proof for DTF2THF_11658
% 3.93/2.11 thf(type_def_5, type, set: ($tType) > $tType).
% 3.93/2.11 thf(type_def_6, type, int: $tType).
% 3.93/2.11 thf(type_def_8, type, real: $tType).
% 3.93/2.11 thf(type_def_9, type, complex: $tType).
% 3.93/2.11 thf(type_def_10, type, bounde2145540817linfun: ($tType * $tType) > $tType).
% 3.93/2.11 thf(type_def_11, type, a: $tType).
% 3.93/2.11 thf(func_def_0, type, bounde2145540817linfun: ($tType * $tType) > $tType).
% 3.93/2.11 thf(func_def_1, type, complex: $tType).
% 3.93/2.11 thf(func_def_2, type, real: $tType).
% 3.93/2.11 thf(func_def_3, type, set: ($tType) > $tType).
% 3.93/2.11 thf(func_def_4, type, int: $tType).
% 3.93/2.11 thf(func_def_5, type, a: $tType).
% 3.93/2.11 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_7, type, zero: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_8, type, top: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_9, type, inner_real_inner: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_10, type, real_V1076094709vector: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_11, type, euclid925273238_space: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_12, type, topolo503727757_space: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_13, type, real_V55928688vector: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_14, type, execut510477386_space: !>[X0: $tType]:($o)).
% 3.93/2.11 thf(func_def_15, type, aff_dim: !>[X0: $tType]:(set @ X0 > int)).
% 3.93/2.11 thf(func_def_16, type, bij_betw: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1 > $o)).
% 3.93/2.11 thf(func_def_17, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 3.93/2.11 thf(func_def_18, type, id: !>[X0: $tType]:(X0 > X0)).
% 3.93/2.11 thf(func_def_19, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 3.93/2.11 thf(func_def_20, type, swap: !>[X0: $tType, X1: $tType]:(X0 > X0 > (X0 > X1) > X0 > X1)).
% 3.93/2.11 thf(func_def_21, type, the_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 3.93/2.11 thf(func_def_22, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.93/2.11 thf(func_def_23, type, hilbert_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 3.93/2.11 thf(func_def_24, type, inner_780170721_inner: !>[X0: $tType]:(X0 > X0 > real)).
% 3.93/2.11 thf(func_def_25, type, linear_adjoint: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1 > X0)).
% 3.93/2.11 thf(func_def_26, type, top_top: !>[X0: $tType]:(X0)).
% 3.93/2.11 thf(func_def_27, type, path_arc: !>[X0: $tType]:((real > X0) > $o)).
% 3.93/2.11 thf(func_def_28, type, path_path: !>[X0: $tType]:((real > X0) > $o)).
% 3.93/2.11 thf(func_def_29, type, path_pathfinish: !>[X0: $tType]:((real > X0) > X0)).
% 3.93/2.11 thf(func_def_30, type, path_pathstart: !>[X0: $tType]:((real > X0) > X0)).
% 3.93/2.11 thf(func_def_31, type, path_rectpath: complex > complex > real > complex).
% 3.93/2.11 thf(func_def_32, type, path_simple_path: !>[X0: $tType]:((real > X0) > $o)).
% 3.93/2.11 thf(func_def_33, type, poinca1076805489pen_R2: !>[X0: $tType]:((X0 > X0) > (X0 > bounde2145540817linfun @ X0 @ X0) > set @ X0 > $o)).
% 3.93/2.11 thf(func_def_34, type, poinca1531302022lex_of: !>[X0: $tType]:(X0 > complex)).
% 3.93/2.11 thf(func_def_35, type, poinca179318102eal_of: !>[X0: $tType]:(complex > X0)).
% 3.93/2.11 thf(func_def_36, type, real_V1632203528linear: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 3.93/2.11 thf(func_def_37, type, real_V1072672427axioms: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 3.93/2.11 thf(func_def_38, type, real_Vector_linear: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 3.93/2.11 thf(func_def_39, type, iso_tu2011167877assist: !>[X0: $tType, X1: $tType]:(((X0 > X0) > X1 > X1) > (X1 > X0) > X1 > (X0 > X0) > X1 > X0 > $o)).
% 3.93/2.11 thf(func_def_40, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.93/2.11 thf(func_def_41, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.93/2.11 thf(func_def_42, type, coplanar: !>[X0: $tType]:(set @ X0 > $o)).
% 3.93/2.11 thf(func_def_43, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.93/2.11 thf(func_def_44, type, c: real > a).
% 3.93/2.11 thf(func_def_60, type, sP0: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 3.93/2.11 thf(func_def_61, type, sK1: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > X0 > X1)).
% 3.93/2.11 thf(func_def_62, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.93/2.11 thf(func_def_63, type, sK3: !>[X0: $tType]:(X0 > (X0 > X0) > X0)).
% 3.93/2.11 thf(func_def_64, type, sK4: !>[X0: $tType]:((X0 > X0) > X0)).
% 3.93/2.11 thf(func_def_65, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > X0 > X1)).
% 3.93/2.11 thf(func_def_66, type, sK6: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > X1) > set @ X0 > X0)).
% 3.93/2.11 thf(func_def_67, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 3.93/2.11 thf(func_def_68, type, sK8: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 3.93/2.11 thf(func_def_69, type, sK9: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X0)).
% 3.93/2.11 thf(func_def_70, type, sK10: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X1)).
% 3.93/2.11 thf(func_def_71, type, sK11: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 3.93/2.11 thf(func_def_72, type, sK12: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 3.93/2.11 thf(func_def_73, type, sK13: !>[X0: $tType, X1: $tType]:(X0 > (X0 > X1) > X0)).
% 3.93/2.11 thf(func_def_74, type, sK14: !>[X0: $tType]:(X0 > X0)).
% 3.93/2.11 thf(func_def_75, type, sK15: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1 > X0)).
% 3.93/2.11 thf(func_def_76, type, sK16: !>[X0: $tType]:(real > X0 > X0)).
% 3.93/2.11 thf(func_def_77, type, sK17: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.93/2.11 thf(func_def_78, type, sK18: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.93/2.11 thf(func_def_79, type, sK19: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 3.93/2.11 thf(func_def_80, type, sK20: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.93/2.11 thf(func_def_81, type, sK21: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 3.93/2.11 thf(func_def_83, type, ph23: !>[X0: $tType]:(X0)).
% 3.93/2.11 thf(f1080,plain,(
% 3.93/2.11 $false),
% 3.93/2.11 inference(avatar_sat_refutation,[status(thm)],[f1063,f1067,f1074])).
% 3.93/2.11 thf(f1074,plain,(
% 3.93/2.11 ~spl22_1 | spl22_3),
% 3.93/2.11 inference(avatar_contradiction_clause,[status(thm)],[f1073])).
% 3.93/2.11 thf(f1073,plain,(
% 3.93/2.11 $false | (~spl22_1 | spl22_3)),
% 3.93/2.11 inference(subsumption_resolution,[status(thm)],[f1072,f1050])).
% 3.93/2.11 thf(f1050,plain,(
% 3.93/2.11 ($true = topolo503727757_space @ a) | ~spl22_1),
% 3.93/2.11 inference(avatar_component_clause,[status(thm)],[f1049])).
% 3.93/2.11 thf(f1049,definition,(
% 3.93/2.11 spl22_1 <=> ($true = topolo503727757_space @ a)),
% 3.93/2.11 introduced(definition,[new_symbols(naming,[spl22_1])],[avatar_definition])).
% 3.93/2.11 thf(f1072,plain,(
% 3.93/2.11 ($true != topolo503727757_space @ a) | spl22_3),
% 3.93/2.11 inference(subsumption_resolution,[status(thm)],[f1071,f1002])).
% 3.93/2.11 thf(f1002,plain,(
% 3.93/2.11 (topolo503727757_space @ complex = $true)),
% 3.93/2.11 inference(cnf_transformation,[status(thm)],[f421])).
% 3.93/2.11 thf(f421,plain,(
% 3.93/2.11 (topolo503727757_space @ complex = $true)),
% 3.93/2.11 inference(fool_elimination,[status(thm)],[f420])).
% 3.93/2.11 thf(f420,plain,(
% 3.93/2.11 topolo503727757_space @ complex),
% 3.93/2.11 inference(rectify,[status(thm)],[f279])).
% 3.93/2.11 thf(f279,axiom,(
% 3.93/2.11 topolo503727757_space @ complex),
% 3.93/2.11 file('/export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2THF_11658.p',tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_9)).
% 3.93/2.11 thf(f1071,plain,(
% 3.93/2.11 (topolo503727757_space @ complex != $true) | ($true != topolo503727757_space @ a) | spl22_3),
% 3.93/2.11 inference(trivial_inequality_removal,[status(thm)],[f1068])).
% 3.93/2.11 thf(f1068,plain,(
% 3.93/2.11 (topolo503727757_space @ complex != $true) | ((poinca1531302022lex_of @ a @ (path_pathfinish @ a @ c)) != (poinca1531302022lex_of @ a @ (path_pathfinish @ a @ c))) | ($true != topolo503727757_space @ a) | spl22_3),
% 3.93/2.11 inference(superposition,[status(thm)],[f1059,f922])).
% 3.93/2.11 thf(f922,plain,(
% 3.93/2.11 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0,X3 : real > X1] : (((X2 @ (path_pathfinish @ X1 @ X3)) = (path_pathfinish @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))) | (topolo503727757_space @ X1 != $true) | (topolo503727757_space @ X0 != $true)) )),
% 3.93/2.11 inference(cnf_transformation,[status(thm)],[f782])).
% 3.93/2.11 thf(f782,plain,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : ((topolo503727757_space @ X0 != $true) | ! [X2 : X1 > X0,X3 : real > X1] : ((X2 @ (path_pathfinish @ X1 @ X3)) = (path_pathfinish @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))) | (topolo503727757_space @ X1 != $true))),
% 3.93/2.11 inference(flattening,[status(thm)],[f781])).
% 3.93/2.11 thf(f781,plain,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : (! [X2 : X1 > X0,X3 : real > X1] : ((X2 @ (path_pathfinish @ X1 @ X3)) = (path_pathfinish @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))) | ((topolo503727757_space @ X1 != $true) | (topolo503727757_space @ X0 != $true)))),
% 3.93/2.11 inference(ennf_transformation,[status(thm)],[f337])).
% 3.93/2.11 thf(f337,plain,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : (((topolo503727757_space @ X1 = $true) & (topolo503727757_space @ X0 = $true)) => ! [X2 : X1 > X0,X3 : real > X1] : ((X2 @ (path_pathfinish @ X1 @ X3)) = (path_pathfinish @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))))),
% 3.93/2.11 inference(fool_elimination,[status(thm)],[f336])).
% 3.93/2.11 thf(f336,plain,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : ((topolo503727757_space @ X0 & topolo503727757_space @ X1) => ! [X2 : X1 > X0,X3 : real > X1] : ((X2 @ (path_pathfinish @ X1 @ X3)) = (path_pathfinish @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))))),
% 3.93/2.11 inference(rectify,[status(thm)],[f5])).
% 3.93/2.11 thf(f5,axiom,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : ((topolo503727757_space @ X0 & topolo503727757_space @ X1) => ! [X6 : X1 > X0,X7 : real > X1] : ((path_pathfinish @ X0 @ (comp @ X1 @ X0 @ real @ X6 @ X7)) = (X6 @ (path_pathfinish @ X1 @ X7))))),
% 3.93/2.11 file('/export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2THF_11658.p',fact_4_pathfinish__compose)).
% 3.93/2.11 thf(f1059,plain,(
% 3.93/2.11 ((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) != (poinca1531302022lex_of @ a @ (path_pathfinish @ a @ c))) | spl22_3),
% 3.93/2.11 inference(avatar_component_clause,[status(thm)],[f1057])).
% 3.93/2.11 thf(f1057,definition,(
% 3.93/2.11 spl22_3 <=> ((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) = (poinca1531302022lex_of @ a @ (path_pathfinish @ a @ c)))),
% 3.93/2.11 introduced(definition,[new_symbols(naming,[spl22_3])],[avatar_definition])).
% 3.93/2.11 thf(f1067,plain,(
% 3.93/2.11 spl22_1),
% 3.93/2.11 inference(avatar_contradiction_clause,[status(thm)],[f1066])).
% 3.93/2.11 thf(f1066,plain,(
% 3.93/2.11 $false | spl22_1),
% 3.93/2.11 inference(subsumption_resolution,[status(thm)],[f1065,f962])).
% 3.93/2.11 thf(f962,plain,(
% 3.93/2.11 (execut510477386_space @ a = $true)),
% 3.93/2.11 inference(cnf_transformation,[status(thm)],[f471])).
% 3.93/2.11 thf(f471,plain,(
% 3.93/2.11 (execut510477386_space @ a = $true)),
% 3.93/2.11 inference(fool_elimination,[status(thm)],[f470])).
% 3.93/2.11 thf(f470,plain,(
% 3.93/2.11 execut510477386_space @ a),
% 3.93/2.11 inference(rectify,[status(thm)],[f284])).
% 3.93/2.11 thf(f284,axiom,(
% 3.93/2.11 execut510477386_space @ a),
% 3.93/2.11 file('/export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2THF_11658.p',tfree_0)).
% 3.93/2.11 thf(f1065,plain,(
% 3.93/2.11 (execut510477386_space @ a != $true) | spl22_1),
% 3.93/2.11 inference(trivial_inequality_removal,[status(thm)],[f1064])).
% 3.93/2.11 thf(f1064,plain,(
% 3.93/2.11 (execut510477386_space @ a != $true) | ($true != $true) | spl22_1),
% 3.93/2.11 inference(superposition,[status(thm)],[f1051,f969])).
% 3.93/2.11 thf(f969,plain,(
% 3.93/2.11 ( ! [X0 : $tType] : ((topolo503727757_space @ X0 = $true) | (execut510477386_space @ X0 != $true)) )),
% 3.93/2.11 inference(cnf_transformation,[status(thm)],[f769])).
% 3.93/2.11 thf(f769,plain,(
% 3.93/2.11 ! [X0 : $tType] : ((execut510477386_space @ X0 != $true) | (topolo503727757_space @ X0 = $true))),
% 3.93/2.11 inference(ennf_transformation,[status(thm)],[f365])).
% 3.93/2.11 thf(f365,plain,(
% 3.93/2.11 ! [X0 : $tType] : ((execut510477386_space @ X0 = $true) => (topolo503727757_space @ X0 = $true))),
% 3.93/2.11 inference(fool_elimination,[status(thm)],[f364])).
% 3.93/2.11 thf(f364,plain,(
% 3.93/2.11 ! [X0 : $tType] : (execut510477386_space @ X0 => topolo503727757_space @ X0)),
% 3.93/2.11 inference(rectify,[status(thm)],[f261])).
% 3.93/2.11 thf(f261,axiom,(
% 3.93/2.11 ! [X0 : $tType] : (execut510477386_space @ X0 => topolo503727757_space @ X0)),
% 3.93/2.11 file('/export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2THF_11658.p',subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Topological__Spaces_Otopological__space)).
% 3.93/2.11 thf(f1051,plain,(
% 3.93/2.11 ($true != topolo503727757_space @ a) | spl22_1),
% 3.93/2.11 inference(avatar_component_clause,[status(thm)],[f1049])).
% 3.93/2.11 thf(f1063,plain,(
% 3.93/2.11 ~spl22_1 | ~spl22_3),
% 3.93/2.11 inference(avatar_split_clause,[status(thm)],[f1062,f1057,f1049])).
% 3.93/2.11 thf(f1062,plain,(
% 3.93/2.11 ((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) != (poinca1531302022lex_of @ a @ (path_pathfinish @ a @ c))) | ($true != topolo503727757_space @ a)),
% 3.93/2.11 inference(forward_demodulation,[status(thm)],[f1061,f1003])).
% 3.93/2.11 thf(f1003,plain,(
% 3.93/2.11 ((path_pathfinish @ a @ c) = (path_pathstart @ a @ c))),
% 3.93/2.11 inference(cnf_transformation,[status(thm)],[f1])).
% 3.93/2.11 thf(f1,axiom,(
% 3.93/2.11 ((path_pathfinish @ a @ c) = (path_pathstart @ a @ c))),
% 3.93/2.11 file('/export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2THF_11658.p',fact_0_assms_I2_J)).
% 3.93/2.11 thf(f1061,plain,(
% 3.93/2.11 ((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) != (poinca1531302022lex_of @ a @ (path_pathstart @ a @ c))) | ($true != topolo503727757_space @ a)),
% 3.93/2.11 inference(subsumption_resolution,[status(thm)],[f1042,f1002])).
% 3.93/2.11 thf(f1042,plain,(
% 3.93/2.11 (topolo503727757_space @ complex != $true) | ($true != topolo503727757_space @ a) | ((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) != (poinca1531302022lex_of @ a @ (path_pathstart @ a @ c)))),
% 3.93/2.11 inference(superposition,[status(thm)],[f937,f899])).
% 3.93/2.11 thf(f899,plain,(
% 3.93/2.11 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0,X3 : real > X1] : (((X2 @ (path_pathstart @ X1 @ X3)) = (path_pathstart @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))) | (topolo503727757_space @ X1 != $true) | (topolo503727757_space @ X0 != $true)) )),
% 3.93/2.11 inference(cnf_transformation,[status(thm)],[f807])).
% 3.93/2.11 thf(f807,plain,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : ((topolo503727757_space @ X1 != $true) | ! [X2 : X1 > X0,X3 : real > X1] : ((X2 @ (path_pathstart @ X1 @ X3)) = (path_pathstart @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))) | (topolo503727757_space @ X0 != $true))),
% 3.93/2.11 inference(flattening,[status(thm)],[f806])).
% 3.93/2.11 thf(f806,plain,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : (! [X2 : X1 > X0,X3 : real > X1] : ((X2 @ (path_pathstart @ X1 @ X3)) = (path_pathstart @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))) | ((topolo503727757_space @ X0 != $true) | (topolo503727757_space @ X1 != $true)))),
% 3.93/2.11 inference(ennf_transformation,[status(thm)],[f459])).
% 3.93/2.11 thf(f459,plain,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : (((topolo503727757_space @ X0 = $true) & (topolo503727757_space @ X1 = $true)) => ! [X2 : X1 > X0,X3 : real > X1] : ((X2 @ (path_pathstart @ X1 @ X3)) = (path_pathstart @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))))),
% 3.93/2.11 inference(fool_elimination,[status(thm)],[f458])).
% 3.93/2.11 thf(f458,plain,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : ((topolo503727757_space @ X1 & topolo503727757_space @ X0) => ! [X2 : X1 > X0,X3 : real > X1] : ((X2 @ (path_pathstart @ X1 @ X3)) = (path_pathstart @ X0 @ (comp @ X1 @ X0 @ real @ X2 @ X3))))),
% 3.93/2.11 inference(rectify,[status(thm)],[f6])).
% 3.93/2.11 thf(f6,axiom,(
% 3.93/2.11 ! [X0 : $tType,X1 : $tType] : ((topolo503727757_space @ X1 & topolo503727757_space @ X0) => ! [X6 : X1 > X0,X7 : real > X1] : ((path_pathstart @ X0 @ (comp @ X1 @ X0 @ real @ X6 @ X7)) = (X6 @ (path_pathstart @ X1 @ X7))))),
% 3.93/2.11 file('/export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2THF_11658.p',fact_5_pathstart__compose)).
% 3.93/2.11 thf(f937,plain,(
% 3.93/2.11 ((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) != (path_pathstart @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)))),
% 3.93/2.11 inference(cnf_transformation,[status(thm)],[f734])).
% 3.93/2.11 thf(f734,plain,(
% 3.93/2.11 ((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) != (path_pathstart @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)))),
% 3.93/2.11 inference(flattening,[status(thm)],[f286])).
% 3.93/2.11 thf(f286,negated_conjecture,(
% 3.93/2.11 ~((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) = (path_pathstart @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)))),
% 3.93/2.11 inference(negated_conjecture,[status(cth)],[f285])).
% 3.93/2.11 thf(f285,conjecture,(
% 3.93/2.11 ((path_pathfinish @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)) = (path_pathstart @ complex @ (comp @ a @ complex @ real @ poinca1531302022lex_of @ a @ c)))),
% 3.93/2.11 file('/export/starexec/sandbox/tmp/tmp.3pDyKuhtmF/DTF2THF_11658.p',conj_0)).
% 3.93/2.11 % SZS output end Proof for DTF2THF_11658
% 3.93/2.11 % (11763)------------------------------
% 3.93/2.11 % (11763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.93/2.11 % (11763)Termination reason: Refutation
% 3.93/2.11
% 3.93/2.11 % (11763)Memory used [KB]: 6652
% 3.93/2.11 % (11763)Time elapsed: 0.045 s
% 3.93/2.11 % (11763)Instructions burned: 70 (million)
% 3.93/2.11 % (11763)------------------------------
% 3.93/2.11 % (11763)------------------------------
% 3.93/2.11 % (11757)Success in time 0.079 s
%------------------------------------------------------------------------------