SET007 Axioms: SET007+828.ax


%------------------------------------------------------------------------------
% File     : SET007+828 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Complex Valued Function's Space
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : cfuncdom [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   61 (   1 unit)
%            Number of atoms       :  403 ( 118 equality)
%            Maximal formula depth :   24 (   9 average)
%            Number of connectives :  421 (  79 ~  ;   2  |; 168  &)
%                                         (   7 <=>; 165 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   0 propositional; 1-4 arity)
%            Number of functors    :   42 (   6 constant; 0-6 arity)
%            Number of variables   :  193 (   0 singleton; 173 !;  20 ?)
%            Maximal term depth    :    6 (   2 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k7_cfuncdom(A))
        & v3_vectsp_1(k7_cfuncdom(A)) ) ) )).

fof(fc2_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k7_cfuncdom(A))
        & v2_group_1(k7_cfuncdom(A))
        & v3_vectsp_1(k7_cfuncdom(A))
        & v6_vectsp_1(k7_cfuncdom(A))
        & v8_vectsp_1(k7_cfuncdom(A)) ) ) )).

fof(rc1_cfuncdom,axiom,(
    ? [A] :
      ( l1_cfuncdom(A)
      & v1_cfuncdom(A) ) )).

fof(rc2_cfuncdom,axiom,(
    ? [A] :
      ( l1_cfuncdom(A)
      & ~ v3_struct_0(A) ) )).

fof(fc3_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k8_cfuncdom(A))
        & v1_cfuncdom(k8_cfuncdom(A)) ) ) )).

fof(fc4_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k8_cfuncdom(A))
        & v2_group_1(k8_cfuncdom(A))
        & v6_vectsp_1(k8_cfuncdom(A))
        & v8_vectsp_1(k8_cfuncdom(A))
        & v1_cfuncdom(k8_cfuncdom(A)) ) ) )).

fof(rc3_cfuncdom,axiom,(
    ? [A] :
      ( l1_cfuncdom(A)
      & ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v1_cfuncdom(A)
      & v2_cfuncdom(A) ) )).

fof(d1_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
            & m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) )
         => ( B = k1_cfuncdom(A)
          <=> ! [C] :
                ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
               => ! [D] :
                    ( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
                   => k1_funcsdom(A,k2_numbers,B,C,D) = k4_funcsdom(k2_numbers,A,k27_binop_2,C,D) ) ) ) ) ) )).

fof(d2_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
            & m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) )
         => ( B = k2_cfuncdom(A)
          <=> ! [C] :
                ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
               => ! [D] :
                    ( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
                   => k1_funcsdom(A,k2_numbers,B,C,D) = k4_funcsdom(k2_numbers,A,k29_binop_2,C,D) ) ) ) ) ) )).

fof(d3_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k2_numbers,k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
            & m2_relset_1(B,k2_zfmisc_1(k2_numbers,k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) )
         => ( B = k3_cfuncdom(A)
          <=> ! [C] :
                ( m1_subset_1(C,k2_numbers)
               => ! [D] :
                    ( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => k8_funct_2(A,k2_numbers,k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),B,k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),C,D)),E) = k9_complex1(C,k8_funct_2(A,k2_numbers,D,E)) ) ) ) ) ) ) )).

fof(d4_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_cfuncdom(A) = k2_funcop_1(A,np__0) ) )).

fof(d5_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k5_cfuncdom(A) = k2_funcop_1(A,k6_complex1) ) )).

fof(t1_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => ! [D] :
                  ( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
                 => ( B = k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,A)
                       => k8_funct_2(A,k2_numbers,B,E) = k8_complex1(k8_funct_2(A,k2_numbers,C,E),k8_funct_2(A,k2_numbers,D,E)) ) ) ) ) ) ) )).

fof(t2_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => ! [D] :
                  ( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
                 => ( B = k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,A)
                       => k8_funct_2(A,k2_numbers,B,E) = k9_complex1(k8_funct_2(A,k2_numbers,C,E),k8_funct_2(A,k2_numbers,D,E)) ) ) ) ) ) ) )).

fof(t3_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => k8_funct_2(A,k2_numbers,k5_cfuncdom(A),B) = k6_complex1 ) ) )).

fof(t4_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => k8_funct_2(A,k2_numbers,k4_cfuncdom(A),B) = np__0 ) ) )).

fof(t5_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_cfuncdom(A) != k5_cfuncdom(A) ) )).

fof(t6_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => ! [D] :
                  ( m1_subset_1(D,k2_numbers)
                 => ( B = k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,C))
                  <=> ! [E] :
                        ( m1_subset_1(E,A)
                       => k8_funct_2(A,k2_numbers,B,E) = k9_complex1(D,k8_funct_2(A,k2_numbers,C,E)) ) ) ) ) ) ) )).

fof(t7_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),B,C) = k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),C,B) ) ) ) )).

fof(t8_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => ! [D] :
                  ( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
                 => k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),B,k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),C,D)) = k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),B,C),D) ) ) ) ) )).

fof(t9_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,C) = k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),C,B) ) ) ) )).

fof(t10_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => ! [D] :
                  ( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
                 => k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),C,D)) = k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,C),D) ) ) ) ) )).

fof(t11_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),k5_cfuncdom(A),B) = B ) ) )).

fof(t12_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),k4_cfuncdom(A),B) = B ) ) )).

fof(t13_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),B,k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),k10_complex1(k6_complex1),B))) = k4_cfuncdom(A) ) ) )).

fof(t14_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),k6_complex1,B)) = B ) ) )).

fof(t15_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m1_subset_1(C,k2_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k2_numbers)
                 => k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),C,k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,B)))) = k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),k9_complex1(C,D),B)) ) ) ) ) )).

fof(t16_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m1_subset_1(C,k2_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k2_numbers)
                 => k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),C,B)),k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,B))) = k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),k8_complex1(C,D),B)) ) ) ) ) )).

fof(t17_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => ! [D] :
                  ( m2_fraenkel(D,A,k2_numbers,k1_fraenkel(A,k2_numbers))
                 => k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),C,D)) = k1_funcsdom(A,k2_numbers,k1_cfuncdom(A),k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,C),k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,D)) ) ) ) ) )).

fof(t18_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k2_numbers,k1_fraenkel(A,k2_numbers))
         => ! [C] :
              ( m2_fraenkel(C,A,k2_numbers,k1_fraenkel(A,k2_numbers))
             => ! [D] :
                  ( m1_subset_1(D,k2_numbers)
                 => k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,B)),C) = k2_funcsdom(A,k2_numbers,k2_numbers,k1_fraenkel(A,k2_numbers),k3_cfuncdom(A),k1_domain_1(k2_numbers,k1_fraenkel(A,k2_numbers),D,k1_funcsdom(A,k2_numbers,k2_cfuncdom(A),B,C))) ) ) ) ) )).

fof(t19_cfuncdom,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ? [C] :
          ( m2_fraenkel(C,B,k2_numbers,k1_fraenkel(B,k2_numbers))
          & ? [D] :
              ( m2_fraenkel(D,B,k2_numbers,k1_fraenkel(B,k2_numbers))
              & ! [E] :
                  ( r2_hidden(E,B)
                 => ( ( E = A
                     => k1_funct_1(C,E) = k6_complex1 )
                    & ( E != A
                     => k1_funct_1(C,E) = np__0 ) ) )
              & ! [E] :
                  ( r2_hidden(E,B)
                 => ( ( E = A
                     => k1_funct_1(D,E) = np__0 )
                    & ( E != A
                     => k1_funct_1(D,E) = k6_complex1 ) ) ) ) ) ) )).

fof(t20_cfuncdom,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
         => ! [E] :
              ( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
             => ( ( r2_hidden(A,C)
                  & r2_hidden(B,C)
                  & ! [F] :
                      ( r2_hidden(F,C)
                     => ( ( F = A
                         => k1_funct_1(D,F) = k6_complex1 )
                        & ( F != A
                         => k1_funct_1(D,F) = np__0 ) ) )
                  & ! [F] :
                      ( r2_hidden(F,C)
                     => ( ( F = A
                         => k1_funct_1(E,F) = np__0 )
                        & ( F != A
                         => k1_funct_1(E,F) = k6_complex1 ) ) ) )
               => ( A = B
                  | ! [F] :
                      ( m1_subset_1(F,k2_numbers)
                     => ! [G] :
                          ( m1_subset_1(G,k2_numbers)
                         => ( k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),F,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,E))) = k4_cfuncdom(C)
                           => ( F = k5_complex1
                              & G = k5_complex1 ) ) ) ) ) ) ) ) ) )).

fof(t21_cfuncdom,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ~ ( r2_hidden(A,C)
          & r2_hidden(B,C)
          & A != B
          & ! [D] :
              ( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
             => ! [E] :
                  ( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
                 => ? [F] :
                      ( m1_subset_1(F,k2_numbers)
                      & ? [G] :
                          ( m1_subset_1(G,k2_numbers)
                          & k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),F,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,E))) = k4_cfuncdom(C)
                          & ~ ( F = np__0
                              & G = np__0 ) ) ) ) ) ) ) )).

fof(t22_cfuncdom,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
         => ! [E] :
              ( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
             => ( ( C = k2_tarski(A,B)
                  & ! [F] :
                      ( r2_hidden(F,C)
                     => ( ( F = A
                         => k1_funct_1(D,F) = k6_complex1 )
                        & ( F != A
                         => k1_funct_1(D,F) = np__0 ) ) )
                  & ! [F] :
                      ( r2_hidden(F,C)
                     => ( ( F = A
                         => k1_funct_1(E,F) = np__0 )
                        & ( F != A
                         => k1_funct_1(E,F) = k6_complex1 ) ) ) )
               => ( A = B
                  | ! [F] :
                      ( m2_fraenkel(F,C,k2_numbers,k1_fraenkel(C,k2_numbers))
                     => ? [G] :
                          ( m1_subset_1(G,k2_numbers)
                          & ? [H] :
                              ( m1_subset_1(H,k2_numbers)
                              & F = k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),H,E))) ) ) ) ) ) ) ) ) )).

fof(t23_cfuncdom,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ~ ( C = k2_tarski(A,B)
          & A != B
          & ! [D] :
              ( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
             => ! [E] :
                  ( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
                 => ? [F] :
                      ( m2_fraenkel(F,C,k2_numbers,k1_fraenkel(C,k2_numbers))
                      & ! [G] :
                          ( m1_subset_1(G,k2_numbers)
                         => ! [H] :
                              ( m1_subset_1(H,k2_numbers)
                             => F != k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),H,E))) ) ) ) ) ) ) ) )).

fof(t24_cfuncdom,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ~ ( C = k2_tarski(A,B)
          & A != B
          & ! [D] :
              ( m2_fraenkel(D,C,k2_numbers,k1_fraenkel(C,k2_numbers))
             => ! [E] :
                  ( m2_fraenkel(E,C,k2_numbers,k1_fraenkel(C,k2_numbers))
                 => ~ ( ! [F] :
                          ( m1_subset_1(F,k2_numbers)
                         => ! [G] :
                              ( m1_subset_1(G,k2_numbers)
                             => ( k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),F,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,E))) = k4_cfuncdom(C)
                               => ( F = np__0
                                  & G = np__0 ) ) ) )
                      & ! [F] :
                          ( m2_fraenkel(F,C,k2_numbers,k1_fraenkel(C,k2_numbers))
                         => ? [G] :
                              ( m1_subset_1(G,k2_numbers)
                              & ? [H] :
                                  ( m1_subset_1(H,k2_numbers)
                                  & F = k1_funcsdom(C,k2_numbers,k1_cfuncdom(C),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),G,D)),k2_funcsdom(C,k2_numbers,k2_numbers,k1_fraenkel(C,k2_numbers),k3_cfuncdom(C),k1_domain_1(k2_numbers,k1_fraenkel(C,k2_numbers),H,E))) ) ) ) ) ) ) ) ) )).

fof(t25_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
        & v3_rlvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
        & v4_rlvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
        & v5_rlvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
        & v6_rlvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
        & v2_clvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)))
        & l1_clvect_1(g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A))) ) ) )).

fof(d6_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k6_cfuncdom(A) = g1_clvect_1(k1_fraenkel(A,k2_numbers),k4_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A)) ) )).

fof(t26_cfuncdom,axiom,(
    ? [A] :
      ( ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v1_clvect_1(A)
      & v2_clvect_1(A)
      & l1_clvect_1(A)
      & ? [B] :
          ( m1_subset_1(B,u1_struct_0(A))
          & ? [C] :
              ( m1_subset_1(C,u1_struct_0(A))
              & ! [D] :
                  ( m1_subset_1(D,k2_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k2_numbers)
                     => ( k4_rlvect_1(A,k1_clvect_1(A,B,D),k1_clvect_1(A,C,E)) = k1_rlvect_1(A)
                       => ( D = np__0
                          & E = np__0 ) ) ) )
              & ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ? [E] :
                      ( m1_subset_1(E,k2_numbers)
                      & ? [F] :
                          ( m1_subset_1(F,k2_numbers)
                          & D = k4_rlvect_1(A,k1_clvect_1(A,B,E),k1_clvect_1(A,C,F)) ) ) ) ) ) ) )).

fof(d7_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k7_cfuncdom(A) = g3_vectsp_1(k1_fraenkel(A,k2_numbers),k1_cfuncdom(A),k2_cfuncdom(A),k5_cfuncdom(A),k4_cfuncdom(A)) ) )).

fof(t27_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k7_cfuncdom(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k7_cfuncdom(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k7_cfuncdom(A)))
                 => ( k2_rlvect_1(k7_cfuncdom(A),B,C) = k2_rlvect_1(k7_cfuncdom(A),C,B)
                    & k2_rlvect_1(k7_cfuncdom(A),k2_rlvect_1(k7_cfuncdom(A),B,C),D) = k2_rlvect_1(k7_cfuncdom(A),B,k2_rlvect_1(k7_cfuncdom(A),C,D))
                    & k2_rlvect_1(k7_cfuncdom(A),B,k1_rlvect_1(k7_cfuncdom(A))) = B
                    & ? [E] :
                        ( m1_subset_1(E,u1_struct_0(k7_cfuncdom(A)))
                        & k2_rlvect_1(k7_cfuncdom(A),B,E) = k1_rlvect_1(k7_cfuncdom(A)) )
                    & k1_group_1(k7_cfuncdom(A),B,C) = k1_group_1(k7_cfuncdom(A),C,B)
                    & k1_group_1(k7_cfuncdom(A),k1_group_1(k7_cfuncdom(A),B,C),D) = k1_group_1(k7_cfuncdom(A),B,k1_group_1(k7_cfuncdom(A),C,D))
                    & k1_group_1(k7_cfuncdom(A),B,k2_group_1(k7_cfuncdom(A))) = B
                    & k1_group_1(k7_cfuncdom(A),k2_group_1(k7_cfuncdom(A)),B) = B
                    & k1_group_1(k7_cfuncdom(A),B,k2_rlvect_1(k7_cfuncdom(A),C,D)) = k2_rlvect_1(k7_cfuncdom(A),k1_group_1(k7_cfuncdom(A),B,C),k1_group_1(k7_cfuncdom(A),B,D))
                    & k1_group_1(k7_cfuncdom(A),k2_rlvect_1(k7_cfuncdom(A),C,D),B) = k2_rlvect_1(k7_cfuncdom(A),k1_group_1(k7_cfuncdom(A),C,B),k1_group_1(k7_cfuncdom(A),D,B)) ) ) ) ) ) )).

fof(t28_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k7_cfuncdom(A))
        & v3_rlvect_1(k7_cfuncdom(A))
        & v4_rlvect_1(k7_cfuncdom(A))
        & v5_rlvect_1(k7_cfuncdom(A))
        & v6_rlvect_1(k7_cfuncdom(A))
        & v4_group_1(k7_cfuncdom(A))
        & v7_group_1(k7_cfuncdom(A))
        & v6_vectsp_1(k7_cfuncdom(A))
        & v7_vectsp_1(k7_cfuncdom(A))
        & v8_vectsp_1(k7_cfuncdom(A))
        & l3_vectsp_1(k7_cfuncdom(A)) ) ) )).

fof(d8_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k8_cfuncdom(A) = g1_cfuncdom(k1_fraenkel(A,k2_numbers),k2_cfuncdom(A),k1_cfuncdom(A),k3_cfuncdom(A),k5_cfuncdom(A),k4_cfuncdom(A)) ) )).

fof(t29_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k8_cfuncdom(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k8_cfuncdom(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k8_cfuncdom(A)))
                 => ! [E] :
                      ( m1_subset_1(E,k2_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k2_numbers)
                         => ( k2_rlvect_1(k8_cfuncdom(A),B,C) = k2_rlvect_1(k8_cfuncdom(A),C,B)
                            & k2_rlvect_1(k8_cfuncdom(A),k2_rlvect_1(k8_cfuncdom(A),B,C),D) = k2_rlvect_1(k8_cfuncdom(A),B,k2_rlvect_1(k8_cfuncdom(A),C,D))
                            & k2_rlvect_1(k8_cfuncdom(A),B,k1_rlvect_1(k8_cfuncdom(A))) = B
                            & ? [G] :
                                ( m1_subset_1(G,u1_struct_0(k8_cfuncdom(A)))
                                & k2_rlvect_1(k8_cfuncdom(A),B,G) = k1_rlvect_1(k8_cfuncdom(A)) )
                            & k1_group_1(k8_cfuncdom(A),B,C) = k1_group_1(k8_cfuncdom(A),C,B)
                            & k1_group_1(k8_cfuncdom(A),k1_group_1(k8_cfuncdom(A),B,C),D) = k1_group_1(k8_cfuncdom(A),B,k1_group_1(k8_cfuncdom(A),C,D))
                            & k1_group_1(k8_cfuncdom(A),B,k2_group_1(k8_cfuncdom(A))) = B
                            & k1_group_1(k8_cfuncdom(A),B,k2_rlvect_1(k8_cfuncdom(A),C,D)) = k2_rlvect_1(k8_cfuncdom(A),k1_group_1(k8_cfuncdom(A),B,C),k1_group_1(k8_cfuncdom(A),B,D))
                            & k1_clvect_1(k8_cfuncdom(A),k1_group_1(k8_cfuncdom(A),B,C),E) = k1_group_1(k8_cfuncdom(A),k1_clvect_1(k8_cfuncdom(A),B,E),C)
                            & k1_clvect_1(k8_cfuncdom(A),k2_rlvect_1(k8_cfuncdom(A),B,C),E) = k2_rlvect_1(k8_cfuncdom(A),k1_clvect_1(k8_cfuncdom(A),B,E),k1_clvect_1(k8_cfuncdom(A),C,E))
                            & k1_clvect_1(k8_cfuncdom(A),B,k8_complex1(E,F)) = k2_rlvect_1(k8_cfuncdom(A),k1_clvect_1(k8_cfuncdom(A),B,E),k1_clvect_1(k8_cfuncdom(A),B,F))
                            & k1_clvect_1(k8_cfuncdom(A),B,k9_complex1(E,F)) = k1_clvect_1(k8_cfuncdom(A),k1_clvect_1(k8_cfuncdom(A),B,F),E) ) ) ) ) ) ) ) )).

fof(d9_cfuncdom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_cfuncdom(A) )
     => ( v2_cfuncdom(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,k2_numbers)
                       => ! [F] :
                            ( m1_subset_1(F,k2_numbers)
                           => ( k1_group_1(A,B,k2_group_1(A)) = B
                              & k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D))
                              & k1_clvect_1(A,k1_group_1(A,B,C),E) = k1_group_1(A,k1_clvect_1(A,B,E),C)
                              & k1_clvect_1(A,k2_rlvect_1(A,B,C),E) = k2_rlvect_1(A,k1_clvect_1(A,B,E),k1_clvect_1(A,C,E))
                              & k1_clvect_1(A,B,k8_complex1(E,F)) = k2_rlvect_1(A,k1_clvect_1(A,B,E),k1_clvect_1(A,B,F))
                              & k1_clvect_1(A,B,k9_complex1(E,F)) = k1_clvect_1(A,k1_clvect_1(A,B,F),E) ) ) ) ) ) ) ) ) )).

fof(t30_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k8_cfuncdom(A))
        & v3_rlvect_1(k8_cfuncdom(A))
        & v4_rlvect_1(k8_cfuncdom(A))
        & v5_rlvect_1(k8_cfuncdom(A))
        & v6_rlvect_1(k8_cfuncdom(A))
        & v4_group_1(k8_cfuncdom(A))
        & v7_group_1(k8_cfuncdom(A))
        & v2_cfuncdom(k8_cfuncdom(A))
        & l1_cfuncdom(k8_cfuncdom(A)) ) ) )).

fof(t31_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k2_group_1(k7_cfuncdom(A)) = k5_cfuncdom(A) ) )).

fof(t32_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k2_group_1(k8_cfuncdom(A)) = k5_cfuncdom(A) ) )).

fof(dt_l1_cfuncdom,axiom,(
    ! [A] :
      ( l1_cfuncdom(A)
     => ( l3_vectsp_1(A)
        & l1_clvect_1(A) ) ) )).

fof(existence_l1_cfuncdom,axiom,(
    ? [A] : l1_cfuncdom(A) )).

fof(abstractness_v1_cfuncdom,axiom,(
    ! [A] :
      ( l1_cfuncdom(A)
     => ( v1_cfuncdom(A)
       => A = g1_cfuncdom(u1_struct_0(A),u1_group_1(A),u1_rlvect_1(A),u1_clvect_1(A),u1_vectsp_1(A),u2_struct_0(A)) ) ) )).

fof(dt_k1_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_funct_1(k1_cfuncdom(A))
        & v1_funct_2(k1_cfuncdom(A),k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
        & m2_relset_1(k1_cfuncdom(A),k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) ) ) )).

fof(dt_k2_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_funct_1(k2_cfuncdom(A))
        & v1_funct_2(k2_cfuncdom(A),k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
        & m2_relset_1(k2_cfuncdom(A),k2_zfmisc_1(k1_fraenkel(A,k2_numbers),k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) ) ) )).

fof(dt_k3_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_funct_1(k3_cfuncdom(A))
        & v1_funct_2(k3_cfuncdom(A),k2_zfmisc_1(k2_numbers,k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers))
        & m2_relset_1(k3_cfuncdom(A),k2_zfmisc_1(k2_numbers,k1_fraenkel(A,k2_numbers)),k1_fraenkel(A,k2_numbers)) ) ) )).

fof(dt_k4_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m2_fraenkel(k4_cfuncdom(A),A,k2_numbers,k1_fraenkel(A,k2_numbers)) ) )).

fof(dt_k5_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m2_fraenkel(k5_cfuncdom(A),A,k2_numbers,k1_fraenkel(A,k2_numbers)) ) )).

fof(dt_k6_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k6_cfuncdom(A))
        & v3_rlvect_1(k6_cfuncdom(A))
        & v4_rlvect_1(k6_cfuncdom(A))
        & v5_rlvect_1(k6_cfuncdom(A))
        & v6_rlvect_1(k6_cfuncdom(A))
        & v1_clvect_1(k6_cfuncdom(A))
        & v2_clvect_1(k6_cfuncdom(A))
        & l1_clvect_1(k6_cfuncdom(A)) ) ) )).

fof(dt_k7_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v3_vectsp_1(k7_cfuncdom(A))
        & l3_vectsp_1(k7_cfuncdom(A)) ) ) )).

fof(dt_k8_cfuncdom,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_cfuncdom(k8_cfuncdom(A))
        & l1_cfuncdom(k8_cfuncdom(A)) ) ) )).

fof(dt_g1_cfuncdom,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(k2_numbers,A),A)
        & m1_relset_1(D,k2_zfmisc_1(k2_numbers,A),A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A) )
     => ( v1_cfuncdom(g1_cfuncdom(A,B,C,D,E,F))
        & l1_cfuncdom(g1_cfuncdom(A,B,C,D,E,F)) ) ) )).

fof(free_g1_cfuncdom,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(k2_numbers,A),A)
        & m1_relset_1(D,k2_zfmisc_1(k2_numbers,A),A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A) )
     => ! [G,H,I,J,K,L] :
          ( g1_cfuncdom(A,B,C,D,E,F) = g1_cfuncdom(G,H,I,J,K,L)
         => ( A = G
            & B = H
            & C = I
            & D = J
            & E = K
            & F = L ) ) ) )).
%------------------------------------------------------------------------------