SET007 Axioms: SET007+138.ax


%------------------------------------------------------------------------------
% File     : SET007+138 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Conjugate, Bounded Complex, and Convergent Complex Sequences
% 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    : comseq_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   59 (   0 unit)
%            Number of atoms       :  479 (  56 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  448 (  28 ~  ;   8  |; 261  &)
%                                         (   6 <=>; 145 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   0 propositional; 1-3 arity)
%            Number of functors    :   33 (   6 constant; 0-4 arity)
%            Number of variables   :  116 (   0 singleton; 106 !;  10 ?)
%            Maximal term depth    :    4 (   1 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_comseq_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k2_numbers)
        & m1_relset_1(B,A,k2_numbers) )
     => ( ~ v1_xboole_0(k1_comseq_2(A,B))
        & v1_relat_1(k1_comseq_2(A,B))
        & v1_funct_1(k1_comseq_2(A,B))
        & v1_funct_2(k1_comseq_2(A,B),A,k2_numbers)
        & v1_partfun1(k1_comseq_2(A,B),A,k2_numbers) ) ) )).

fof(rc1_comseq_2,axiom,(
    ? [A] :
      ( m1_relset_1(A,k5_numbers,k2_numbers)
      & ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k2_numbers)
      & v1_partfun1(A,k5_numbers,k2_numbers)
      & v1_comseq_2(A) ) )).

fof(rc2_comseq_2,axiom,(
    ? [A] :
      ( m1_relset_1(A,k5_numbers,k2_numbers)
      & ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k2_numbers)
      & v1_partfun1(A,k5_numbers,k2_numbers)
      & v2_comseq_2(A) ) )).

fof(fc2_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & v2_comseq_2(A)
        & m1_relset_1(A,k5_numbers,k2_numbers) )
     => ( ~ v1_xboole_0(k9_comseq_1(k5_numbers,A))
        & v1_relat_1(k9_comseq_1(k5_numbers,A))
        & v1_funct_1(k9_comseq_1(k5_numbers,A))
        & v1_funct_2(k9_comseq_1(k5_numbers,A),k5_numbers,k1_numbers)
        & v4_seq_2(k9_comseq_1(k5_numbers,A))
        & v1_partfun1(k9_comseq_1(k5_numbers,A),k5_numbers,k1_numbers) ) ) )).

fof(fc3_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & v2_comseq_2(A)
        & m1_relset_1(A,k5_numbers,k2_numbers) )
     => ( ~ v1_xboole_0(k1_comseq_2(k5_numbers,A))
        & v1_relat_1(k1_comseq_2(k5_numbers,A))
        & v1_funct_1(k1_comseq_2(k5_numbers,A))
        & v1_funct_2(k1_comseq_2(k5_numbers,A),k5_numbers,k2_numbers)
        & v1_partfun1(k1_comseq_2(k5_numbers,A),k5_numbers,k2_numbers)
        & v2_comseq_2(k1_comseq_2(k5_numbers,A)) ) ) )).

fof(cc1_comseq_2,axiom,(
    ! [A] :
      ( m1_relset_1(A,k5_numbers,k2_numbers)
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & v2_comseq_2(A) )
       => ( ~ v1_xboole_0(A)
          & v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & v1_partfun1(A,k5_numbers,k2_numbers)
          & v1_comseq_2(A) ) ) ) )).

fof(cc2_comseq_2,axiom,(
    ! [A] :
      ( m1_relset_1(A,k5_numbers,k2_numbers)
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & ~ v1_comseq_2(A) )
       => ( ~ v1_xboole_0(A)
          & v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & v1_partfun1(A,k5_numbers,k2_numbers)
          & ~ v2_comseq_2(A) ) ) ) )).

fof(t1_comseq_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => ~ ( A != k5_complex1
              & B != k5_complex1
              & k17_complex1(k11_complex1(k12_complex1(A),k12_complex1(B))) != k6_real_1(k17_complex1(k11_complex1(A,B)),k4_real_1(k17_complex1(A),k17_complex1(B))) ) ) ) )).

fof(t2_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ? [C] :
              ( m1_subset_1(C,k1_numbers)
              & ~ r1_xreal_0(C,np__0)
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( r1_xreal_0(D,B)
                      & r1_xreal_0(C,k17_complex1(k1_comseq_1(A,D))) ) ) ) ) ) )).

fof(d1_comseq_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k2_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k2_numbers) )
             => ( C = k1_comseq_2(A,B)
              <=> ( k4_relset_1(A,k2_numbers,C) = k4_relset_1(A,k2_numbers,B)
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => ( r2_hidden(D,k4_relset_1(A,k2_numbers,C))
                       => k1_funct_1(C,D) = k15_complex1(k4_finseq_4(A,k2_numbers,B,D)) ) ) ) ) ) ) ) )).

fof(d2_comseq_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k2_numbers)
            & m2_relset_1(B,A,k2_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k2_numbers) )
             => ( C = k1_comseq_2(A,B)
              <=> ( k4_relset_1(A,k2_numbers,C) = A
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => k1_funct_1(C,D) = k15_complex1(k8_funct_2(A,k2_numbers,B,D)) ) ) ) ) ) ) )).

fof(t3_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v1_comseq_1(A)
       => v1_comseq_1(k1_comseq_2(k5_numbers,A)) ) ) )).

fof(t4_comseq_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => k1_comseq_2(k5_numbers,k4_comseq_1(k5_numbers,B,A)) = k4_comseq_1(k5_numbers,k1_comseq_2(k5_numbers,B),k15_complex1(A)) ) ) )).

fof(t5_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => k1_comseq_2(k5_numbers,k3_comseq_1(k5_numbers,A,B)) = k3_comseq_1(k5_numbers,k1_comseq_2(k5_numbers,A),k1_comseq_2(k5_numbers,B)) ) ) )).

fof(t6_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => k7_comseq_1(k1_comseq_2(k5_numbers,A)) = k1_comseq_2(k5_numbers,k7_comseq_1(A)) ) )).

fof(t7_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => k1_comseq_2(k5_numbers,k8_comseq_1(A,B)) = k8_comseq_1(k1_comseq_2(k5_numbers,A),k1_comseq_2(k5_numbers,B)) ) ) )).

fof(d3_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v1_comseq_2(A)
      <=> ? [B] :
            ( m1_subset_1(B,k1_numbers)
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ r1_xreal_0(B,k17_complex1(k1_comseq_1(A,C))) ) ) ) ) )).

fof(t8_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v1_comseq_2(A)
      <=> ? [B] :
            ( m1_subset_1(B,k1_numbers)
            & ~ r1_xreal_0(B,np__0)
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ r1_xreal_0(B,k17_complex1(k1_comseq_1(A,C))) ) ) ) ) )).

fof(d4_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v2_comseq_2(A)
      <=> ? [B] :
            ( m1_subset_1(B,k2_numbers)
            & ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,np__0)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ? [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & r1_xreal_0(D,E)
                            & r1_xreal_0(C,k17_complex1(k11_complex1(k1_comseq_1(A,E),B))) ) ) ) ) ) ) ) )).

fof(d5_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v2_comseq_2(A)
       => ! [B] :
            ( m1_subset_1(B,k2_numbers)
           => ( B = k2_comseq_2(A)
            <=> ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ? [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                              & r1_xreal_0(D,E)
                              & r1_xreal_0(C,k17_complex1(k11_complex1(k1_comseq_1(A,E),B))) ) ) ) ) ) ) ) ) )).

fof(t9_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( ? [B] :
            ( m1_subset_1(B,k2_numbers)
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k1_comseq_1(A,C) = B ) )
       => v2_comseq_2(A) ) ) )).

fof(t10_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k1_comseq_1(A,C) = B )
           => k2_comseq_2(A) = B ) ) ) )).

fof(t11_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v2_comseq_2(A)
       => k2_seq_2(k9_comseq_1(k5_numbers,A)) = k17_complex1(k2_comseq_2(A)) ) ) )).

fof(t12_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v2_comseq_2(A)
       => k2_comseq_2(k1_comseq_2(k5_numbers,A)) = k15_complex1(k2_comseq_2(A)) ) ) )).

fof(t13_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => v2_comseq_2(k2_comseq_1(k5_numbers,A,B)) ) ) ) )).

fof(t14_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => k2_comseq_2(k2_comseq_1(k5_numbers,A,B)) = k8_complex1(k2_comseq_2(A),k2_comseq_2(B)) ) ) ) )).

fof(t15_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => k2_seq_2(k9_comseq_1(k5_numbers,k2_comseq_1(k5_numbers,A,B))) = k17_complex1(k8_complex1(k2_comseq_2(A),k2_comseq_2(B))) ) ) ) )).

fof(t16_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => k2_comseq_2(k1_comseq_2(k5_numbers,k2_comseq_1(k5_numbers,A,B))) = k8_complex1(k15_complex1(k2_comseq_2(A)),k15_complex1(k2_comseq_2(B))) ) ) ) )).

fof(t17_comseq_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( v2_comseq_2(B)
           => v2_comseq_2(k4_comseq_1(k5_numbers,B,A)) ) ) ) )).

fof(t18_comseq_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( v2_comseq_2(B)
           => k2_comseq_2(k4_comseq_1(k5_numbers,B,A)) = k9_complex1(A,k2_comseq_2(B)) ) ) ) )).

fof(t19_comseq_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( v2_comseq_2(B)
           => k2_seq_2(k9_comseq_1(k5_numbers,k4_comseq_1(k5_numbers,B,A))) = k4_real_1(k17_complex1(A),k17_complex1(k2_comseq_2(B))) ) ) ) )).

fof(t20_comseq_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( v2_comseq_2(B)
           => k2_comseq_2(k1_comseq_2(k5_numbers,k4_comseq_1(k5_numbers,B,A))) = k9_complex1(k15_complex1(A),k15_complex1(k2_comseq_2(B))) ) ) ) )).

fof(t21_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v2_comseq_2(A)
       => v2_comseq_2(k5_comseq_1(k5_numbers,A)) ) ) )).

fof(t22_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v2_comseq_2(A)
       => k2_comseq_2(k5_comseq_1(k5_numbers,A)) = k10_complex1(k2_comseq_2(A)) ) ) )).

fof(t23_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v2_comseq_2(A)
       => k2_seq_2(k9_comseq_1(k5_numbers,k5_comseq_1(k5_numbers,A))) = k17_complex1(k2_comseq_2(A)) ) ) )).

fof(t24_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v2_comseq_2(A)
       => k2_comseq_2(k1_comseq_2(k5_numbers,k5_comseq_1(k5_numbers,A))) = k10_complex1(k15_complex1(k2_comseq_2(A))) ) ) )).

fof(t25_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => v2_comseq_2(k6_comseq_1(k5_numbers,A,B)) ) ) ) )).

fof(t26_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => k2_comseq_2(k6_comseq_1(k5_numbers,A,B)) = k11_complex1(k2_comseq_2(A),k2_comseq_2(B)) ) ) ) )).

fof(t27_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => k2_seq_2(k9_comseq_1(k5_numbers,k6_comseq_1(k5_numbers,A,B))) = k17_complex1(k11_complex1(k2_comseq_2(A),k2_comseq_2(B))) ) ) ) )).

fof(t28_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => k2_comseq_2(k1_comseq_2(k5_numbers,k6_comseq_1(k5_numbers,A,B))) = k11_complex1(k15_complex1(k2_comseq_2(A)),k15_complex1(k2_comseq_2(B))) ) ) ) )).

fof(t29_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v1_funct_1(A)
              & v1_funct_2(A,k5_numbers,k2_numbers)
              & v2_comseq_2(A)
              & m2_relset_1(A,k5_numbers,k2_numbers)
              & v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k2_numbers)
              & v2_comseq_2(B)
              & m2_relset_1(B,k5_numbers,k2_numbers) )
           => v2_comseq_2(k3_comseq_1(k5_numbers,A,B)) ) ) ) )).

fof(t30_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v1_funct_1(A)
              & v1_funct_2(A,k5_numbers,k2_numbers)
              & v2_comseq_2(A)
              & m2_relset_1(A,k5_numbers,k2_numbers)
              & v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k2_numbers)
              & v2_comseq_2(B)
              & m2_relset_1(B,k5_numbers,k2_numbers) )
           => k2_comseq_2(k3_comseq_1(k5_numbers,A,B)) = k9_complex1(k2_comseq_2(A),k2_comseq_2(B)) ) ) ) )).

fof(t31_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => k2_seq_2(k9_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B))) = k4_real_1(k17_complex1(k2_comseq_2(A)),k17_complex1(k2_comseq_2(B))) ) ) ) )).

fof(t32_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B) )
           => k2_comseq_2(k1_comseq_2(k5_numbers,k3_comseq_1(k5_numbers,A,B))) = k9_complex1(k15_complex1(k2_comseq_2(A)),k15_complex1(k2_comseq_2(B))) ) ) ) )).

fof(t33_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ~ ( v2_comseq_2(A)
          & k2_comseq_2(A) != k5_complex1
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ? [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                  & r1_xreal_0(B,C)
                  & r1_xreal_0(k17_complex1(k1_comseq_1(A,C)),k6_real_1(k17_complex1(k2_comseq_2(A)),np__2)) ) ) ) ) )).

fof(t34_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( ( v2_comseq_2(A)
          & v1_comseq_1(A) )
       => ( k2_comseq_2(A) = k5_complex1
          | v2_comseq_2(k7_comseq_1(A)) ) ) ) )).

fof(t35_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( ( v2_comseq_2(A)
          & v1_comseq_1(A) )
       => ( k2_comseq_2(A) = k5_complex1
          | k2_comseq_2(k7_comseq_1(A)) = k12_complex1(k2_comseq_2(A)) ) ) ) )).

fof(t36_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( ( v2_comseq_2(A)
          & v1_comseq_1(A) )
       => ( k2_comseq_2(A) = k5_complex1
          | k2_seq_2(k9_comseq_1(k5_numbers,k7_comseq_1(A))) = k2_real_1(k17_complex1(k2_comseq_2(A))) ) ) ) )).

fof(t37_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( ( v2_comseq_2(A)
          & v1_comseq_1(A) )
       => ( k2_comseq_2(A) = k5_complex1
          | k2_comseq_2(k1_comseq_2(k5_numbers,k7_comseq_1(A))) = k12_complex1(k15_complex1(k2_comseq_2(A))) ) ) ) )).

fof(t38_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B)
              & v1_comseq_1(B) )
           => ( k2_comseq_2(B) = k5_complex1
              | v2_comseq_2(k8_comseq_1(A,B)) ) ) ) ) )).

fof(t39_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B)
              & v1_comseq_1(B) )
           => ( k2_comseq_2(B) = k5_complex1
              | k2_comseq_2(k8_comseq_1(A,B)) = k13_complex1(k2_comseq_2(A),k2_comseq_2(B)) ) ) ) ) )).

fof(t40_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B)
              & v1_comseq_1(B) )
           => ( k2_comseq_2(B) = k5_complex1
              | k2_seq_2(k9_comseq_1(k5_numbers,k8_comseq_1(A,B))) = k6_real_1(k17_complex1(k2_comseq_2(A)),k17_complex1(k2_comseq_2(B))) ) ) ) ) )).

fof(t41_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v2_comseq_2(B)
              & v1_comseq_1(B) )
           => ( k2_comseq_2(B) = k5_complex1
              | k2_comseq_2(k1_comseq_2(k5_numbers,k8_comseq_1(A,B))) = k13_complex1(k15_complex1(k2_comseq_2(A)),k15_complex1(k2_comseq_2(B))) ) ) ) ) )).

fof(t42_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v1_comseq_2(B)
              & k2_comseq_2(A) = k5_complex1 )
           => v2_comseq_2(k3_comseq_1(k5_numbers,A,B)) ) ) ) )).

fof(t43_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v1_comseq_2(B)
              & k2_comseq_2(A) = k5_complex1 )
           => k2_comseq_2(k3_comseq_1(k5_numbers,A,B)) = k5_complex1 ) ) ) )).

fof(t44_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v1_comseq_2(B)
              & k2_comseq_2(A) = k5_complex1 )
           => k2_seq_2(k9_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B))) = np__0 ) ) ) )).

fof(t45_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ( ( v2_comseq_2(A)
              & v1_comseq_2(B)
              & k2_comseq_2(A) = k5_complex1 )
           => k2_comseq_2(k1_comseq_2(k5_numbers,k3_comseq_1(k5_numbers,A,B))) = k5_complex1 ) ) ) )).

fof(dt_k1_comseq_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers) )
     => ( v1_funct_1(k1_comseq_2(A,B))
        & m2_relset_1(k1_comseq_2(A,B),A,k2_numbers) ) ) )).

fof(dt_k2_comseq_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m1_relset_1(A,k5_numbers,k2_numbers) )
     => m1_subset_1(k2_comseq_2(A),k2_numbers) ) )).
%------------------------------------------------------------------------------