SET007 Axioms: SET007+128.ax


%------------------------------------------------------------------------------
% File     : SET007+128 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : 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_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   87 (   3 unit)
%            Number of atoms       :  703 (  89 equality)
%            Maximal formula depth :   14 (   8 average)
%            Number of connectives :  650 (  34 ~  ;   2  |; 386  &)
%                                         (  17 <=>; 211 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-3 arity)
%            Number of functors    :   35 (   5 constant; 0-4 arity)
%            Number of variables   :  220 (   0 singleton; 217 !;   3 ?)
%            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(rc1_comseq_1,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_1(A) ) )).

fof(fc1_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k2_numbers)
        & m1_relset_1(B,A,k2_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k2_numbers)
        & m1_relset_1(C,A,k2_numbers) )
     => ( ~ v1_xboole_0(k2_comseq_1(A,B,C))
        & v1_relat_1(k2_comseq_1(A,B,C))
        & v1_funct_1(k2_comseq_1(A,B,C))
        & v1_funct_2(k2_comseq_1(A,B,C),A,k2_numbers)
        & v1_partfun1(k2_comseq_1(A,B,C),A,k2_numbers) ) ) )).

fof(fc2_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k2_numbers)
        & m1_relset_1(B,A,k2_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k2_numbers)
        & m1_relset_1(C,A,k2_numbers) )
     => ( ~ v1_xboole_0(k3_comseq_1(A,B,C))
        & v1_relat_1(k3_comseq_1(A,B,C))
        & v1_funct_1(k3_comseq_1(A,B,C))
        & v1_funct_2(k3_comseq_1(A,B,C),A,k2_numbers)
        & v1_partfun1(k3_comseq_1(A,B,C),A,k2_numbers) ) ) )).

fof(fc3_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k2_numbers)
        & m1_relset_1(B,A,k2_numbers)
        & v1_xcmplx_0(C) )
     => ( ~ v1_xboole_0(k4_comseq_1(A,B,C))
        & v1_relat_1(k4_comseq_1(A,B,C))
        & v1_funct_1(k4_comseq_1(A,B,C))
        & v1_funct_2(k4_comseq_1(A,B,C),A,k2_numbers)
        & v1_partfun1(k4_comseq_1(A,B,C),A,k2_numbers) ) ) )).

fof(fc4_comseq_1,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(k5_comseq_1(A,B))
        & v1_relat_1(k5_comseq_1(A,B))
        & v1_funct_1(k5_comseq_1(A,B))
        & v1_funct_2(k5_comseq_1(A,B),A,k2_numbers)
        & v1_partfun1(k5_comseq_1(A,B),A,k2_numbers) ) ) )).

fof(fc5_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k2_numbers)
        & m1_relset_1(B,A,k2_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k2_numbers)
        & m1_relset_1(C,A,k2_numbers) )
     => ( ~ v1_xboole_0(k6_comseq_1(A,B,C))
        & v1_relat_1(k6_comseq_1(A,B,C))
        & v1_funct_1(k6_comseq_1(A,B,C))
        & v1_funct_2(k6_comseq_1(A,B,C),A,k2_numbers)
        & v1_partfun1(k6_comseq_1(A,B,C),A,k2_numbers) ) ) )).

fof(fc6_comseq_1,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(k9_comseq_1(A,B))
        & v1_relat_1(k9_comseq_1(A,B))
        & v1_funct_1(k9_comseq_1(A,B))
        & v1_funct_2(k9_comseq_1(A,B),A,k1_numbers)
        & v1_partfun1(k9_comseq_1(A,B),A,k1_numbers) ) ) )).

fof(t1_comseq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & m2_relset_1(A,k5_numbers,k2_numbers) )
      <=> ( k1_relat_1(A) = k5_numbers
          & ! [B] :
              ( r2_hidden(B,k5_numbers)
             => m1_subset_1(k1_funct_1(A,B),k2_numbers) ) ) ) ) )).

fof(t2_comseq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & m2_relset_1(A,k5_numbers,k2_numbers) )
      <=> ( k1_relat_1(A) = k5_numbers
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => m1_subset_1(k1_funct_1(A,B),k2_numbers) ) ) ) ) )).

fof(d1_comseq_1,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)
      <=> r1_tarski(k5_relset_1(k5_numbers,k2_numbers,A),k4_xboole_0(k2_numbers,k1_tarski(k5_complex1))) ) ) )).

fof(t3_comseq_1,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)
      <=> ! [B] :
            ~ ( r2_hidden(B,k5_numbers)
              & k1_funct_1(A,B) = k5_complex1 ) ) ) )).

fof(t4_comseq_1,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)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k1_comseq_1(A,B) != k5_complex1 ) ) ) )).

fof(t5_comseq_1,axiom,(
    $true )).

fof(t6_comseq_1,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) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k1_comseq_1(A,C) = k1_comseq_1(B,C) )
           => A = B ) ) ) )).

fof(t7_comseq_1,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)
          & k5_relset_1(k5_numbers,k2_numbers,B) = k1_tarski(A) ) ) )).

fof(d2_comseq_1,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) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k2_numbers) )
                 => ( D = k2_comseq_1(A,B,C)
                  <=> ( k4_relset_1(A,k2_numbers,D) = k5_subset_1(A,k4_relset_1(A,k2_numbers,B),k4_relset_1(A,k2_numbers,C))
                      & ! [E] :
                          ( m1_subset_1(E,A)
                         => ( r2_hidden(E,k4_relset_1(A,k2_numbers,D))
                           => k1_funct_1(D,E) = k8_complex1(k4_finseq_4(A,k2_numbers,B,E),k4_finseq_4(A,k2_numbers,C,E)) ) ) ) ) ) ) ) ) )).

fof(d3_comseq_1,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) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k2_numbers) )
                 => ( D = k3_comseq_1(A,B,C)
                  <=> ( k4_relset_1(A,k2_numbers,D) = k5_subset_1(A,k4_relset_1(A,k2_numbers,B),k4_relset_1(A,k2_numbers,C))
                      & ! [E] :
                          ( m1_subset_1(E,A)
                         => ( r2_hidden(E,k4_relset_1(A,k2_numbers,D))
                           => k1_funct_1(D,E) = k9_complex1(k4_finseq_4(A,k2_numbers,B,E),k4_finseq_4(A,k2_numbers,C,E)) ) ) ) ) ) ) ) ) )).

fof(d4_comseq_1,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)
                & v1_funct_2(C,A,k2_numbers)
                & m2_relset_1(C,A,k2_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k2_numbers) )
                 => ( D = k2_comseq_1(A,B,C)
                  <=> ( k4_relset_1(A,k2_numbers,D) = A
                      & ! [E] :
                          ( m1_subset_1(E,A)
                         => k1_funct_1(D,E) = k8_complex1(k8_funct_2(A,k2_numbers,B,E),k8_funct_2(A,k2_numbers,C,E)) ) ) ) ) ) ) ) )).

fof(d5_comseq_1,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)
                & v1_funct_2(C,A,k2_numbers)
                & m2_relset_1(C,A,k2_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k2_numbers) )
                 => ( D = k3_comseq_1(A,B,C)
                  <=> ( k4_relset_1(A,k2_numbers,D) = A
                      & ! [E] :
                          ( m1_subset_1(E,A)
                         => k1_funct_1(D,E) = k9_complex1(k8_funct_2(A,k2_numbers,B,E),k8_funct_2(A,k2_numbers,C,E)) ) ) ) ) ) ) ) )).

fof(d6_comseq_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k2_numbers) )
         => ! [C] :
              ( v1_xcmplx_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k2_numbers) )
                 => ( D = k4_comseq_1(A,B,C)
                  <=> ( k4_relset_1(A,k2_numbers,D) = k4_relset_1(A,k2_numbers,B)
                      & ! [E] :
                          ( m1_subset_1(E,A)
                         => ( r2_hidden(E,k4_relset_1(A,k2_numbers,D))
                           => k1_funct_1(D,E) = k3_xcmplx_0(C,k4_finseq_4(A,k2_numbers,B,E)) ) ) ) ) ) ) ) ) )).

fof(d7_comseq_1,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_xcmplx_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k2_numbers) )
                 => ( D = k4_comseq_1(A,B,C)
                  <=> ( k4_relset_1(A,k2_numbers,D) = A
                      & ! [E] :
                          ( m1_subset_1(E,A)
                         => k1_funct_1(D,E) = k3_xcmplx_0(C,k8_funct_2(A,k2_numbers,B,E)) ) ) ) ) ) ) ) )).

fof(d8_comseq_1,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 = k5_comseq_1(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) = k10_complex1(k4_finseq_4(A,k2_numbers,B,D)) ) ) ) ) ) ) ) )).

fof(d9_comseq_1,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 = k5_comseq_1(A,B)
              <=> ( k4_relset_1(A,k2_numbers,C) = A
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => k1_funct_1(C,D) = k10_complex1(k8_funct_2(A,k2_numbers,B,D)) ) ) ) ) ) ) )).

fof(d10_comseq_1,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) )
             => k6_comseq_1(A,B,C) = k2_comseq_1(A,B,k5_comseq_1(A,C)) ) ) ) )).

fof(d11_comseq_1,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) )
         => ( B = k7_comseq_1(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k1_comseq_1(B,C) = k12_complex1(k1_comseq_1(A,C)) ) ) ) ) )).

fof(d12_comseq_1,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) )
         => k8_comseq_1(A,B) = k3_comseq_1(k5_numbers,A,k7_comseq_1(B)) ) ) )).

fof(d13_comseq_1,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,k1_numbers) )
             => ( C = k9_comseq_1(A,B)
              <=> ( k4_relset_1(A,k1_numbers,C) = k4_relset_1(A,k2_numbers,B)
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => ( r2_hidden(D,k4_relset_1(A,k1_numbers,C))
                       => k2_seq_1(A,k1_numbers,C,D) = k17_complex1(k4_finseq_4(A,k2_numbers,B,D)) ) ) ) ) ) ) ) )).

fof(d14_comseq_1,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,k1_numbers) )
             => ( C = k9_comseq_1(A,B)
              <=> ( k4_relset_1(A,k1_numbers,C) = A
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => k2_seq_1(A,k1_numbers,C,D) = k17_complex1(k8_funct_2(A,k2_numbers,B,D)) ) ) ) ) ) ) )).

fof(t8_comseq_1,axiom,(
    $true )).

fof(t9_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k2_comseq_1(k5_numbers,k2_comseq_1(k5_numbers,A,B),C) = k2_comseq_1(k5_numbers,A,k2_comseq_1(k5_numbers,B,C)) ) ) ) )).

fof(t10_comseq_1,axiom,(
    $true )).

fof(t11_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k3_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B),C) = k3_comseq_1(k5_numbers,A,k3_comseq_1(k5_numbers,B,C)) ) ) ) )).

fof(t12_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k3_comseq_1(k5_numbers,k2_comseq_1(k5_numbers,A,B),C) = k2_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,C),k3_comseq_1(k5_numbers,B,C)) ) ) ) )).

fof(t13_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k3_comseq_1(k5_numbers,A,k2_comseq_1(k5_numbers,B,C)) = k2_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B),k3_comseq_1(k5_numbers,A,C)) ) ) ) )).

fof(t14_comseq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => k5_comseq_1(k5_numbers,A) = k4_comseq_1(k5_numbers,A,k10_complex1(k6_complex1)) ) )).

fof(t15_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k4_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,B,C),A) = k3_comseq_1(k5_numbers,k4_comseq_1(k5_numbers,B,A),C) ) ) ) )).

fof(t16_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k4_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,B,C),A) = k3_comseq_1(k5_numbers,B,k4_comseq_1(k5_numbers,C,A)) ) ) ) )).

fof(t17_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k3_comseq_1(k5_numbers,k6_comseq_1(k5_numbers,A,B),C) = k6_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,C),k3_comseq_1(k5_numbers,B,C)) ) ) ) )).

fof(t18_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k6_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B),k3_comseq_1(k5_numbers,A,C)) = k3_comseq_1(k5_numbers,A,k6_comseq_1(k5_numbers,B,C)) ) ) ) )).

fof(t19_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k4_comseq_1(k5_numbers,k2_comseq_1(k5_numbers,B,C),A) = k2_comseq_1(k5_numbers,k4_comseq_1(k5_numbers,B,A),k4_comseq_1(k5_numbers,C,A)) ) ) ) )).

fof(t20_comseq_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k4_comseq_1(k5_numbers,C,k9_complex1(A,B)) = k4_comseq_1(k5_numbers,k4_comseq_1(k5_numbers,C,B),A) ) ) ) )).

fof(t21_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k4_comseq_1(k5_numbers,k6_comseq_1(k5_numbers,B,C),A) = k6_comseq_1(k5_numbers,k4_comseq_1(k5_numbers,B,A),k4_comseq_1(k5_numbers,C,A)) ) ) ) )).

fof(t22_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k4_comseq_1(k5_numbers,k8_comseq_1(B,C),A) = k8_comseq_1(k4_comseq_1(k5_numbers,B,A),C) ) ) ) )).

fof(t23_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k6_comseq_1(k5_numbers,A,k2_comseq_1(k5_numbers,B,C)) = k6_comseq_1(k5_numbers,k6_comseq_1(k5_numbers,A,B),C) ) ) ) )).

fof(t24_comseq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => k4_comseq_1(k5_numbers,A,k6_complex1) = A ) )).

fof(t25_comseq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => k5_comseq_1(k5_numbers,k5_comseq_1(k5_numbers,A)) = A ) )).

fof(t26_comseq_1,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) )
         => k6_comseq_1(k5_numbers,A,k5_comseq_1(k5_numbers,B)) = k2_comseq_1(k5_numbers,A,B) ) ) )).

fof(t27_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k6_comseq_1(k5_numbers,A,k6_comseq_1(k5_numbers,B,C)) = k2_comseq_1(k5_numbers,k6_comseq_1(k5_numbers,A,B),C) ) ) ) )).

fof(t28_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k2_comseq_1(k5_numbers,A,k6_comseq_1(k5_numbers,B,C)) = k6_comseq_1(k5_numbers,k2_comseq_1(k5_numbers,A,B),C) ) ) ) )).

fof(t29_comseq_1,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) )
         => ( k3_comseq_1(k5_numbers,k5_comseq_1(k5_numbers,A),B) = k5_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B))
            & k3_comseq_1(k5_numbers,A,k5_comseq_1(k5_numbers,B)) = k5_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B)) ) ) ) )).

fof(t30_comseq_1,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(k7_comseq_1(A)) ) ) )).

fof(t31_comseq_1,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(k7_comseq_1(A)) = A ) )).

fof(t32_comseq_1,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_comseq_1(A)
              & v1_comseq_1(B) )
          <=> v1_comseq_1(k3_comseq_1(k5_numbers,A,B)) ) ) ) )).

fof(t33_comseq_1,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_comseq_1(A)
              & v1_comseq_1(B) )
           => k3_comseq_1(k5_numbers,k7_comseq_1(A),k7_comseq_1(B)) = k7_comseq_1(k3_comseq_1(k5_numbers,A,B)) ) ) ) )).

fof(t34_comseq_1,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_comseq_1(A)
           => k3_comseq_1(k5_numbers,k8_comseq_1(B,A),A) = B ) ) ) )).

fof(t35_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k2_numbers)
                    & m2_relset_1(D,k5_numbers,k2_numbers) )
                 => ( ( v1_comseq_1(A)
                      & v1_comseq_1(B) )
                   => k3_comseq_1(k5_numbers,k8_comseq_1(C,A),k8_comseq_1(D,B)) = k8_comseq_1(k3_comseq_1(k5_numbers,C,D),k3_comseq_1(k5_numbers,A,B)) ) ) ) ) ) )).

fof(t36_comseq_1,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_comseq_1(A)
              & v1_comseq_1(B) )
           => v1_comseq_1(k8_comseq_1(A,B)) ) ) ) )).

fof(t37_comseq_1,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_comseq_1(A)
              & v1_comseq_1(B) )
           => k7_comseq_1(k8_comseq_1(A,B)) = k8_comseq_1(B,A) ) ) ) )).

fof(t38_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => k3_comseq_1(k5_numbers,A,k8_comseq_1(B,C)) = k8_comseq_1(k3_comseq_1(k5_numbers,A,B),C) ) ) ) )).

fof(t39_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => ( ( v1_comseq_1(A)
                  & v1_comseq_1(B) )
               => k8_comseq_1(C,k8_comseq_1(A,B)) = k8_comseq_1(k3_comseq_1(k5_numbers,C,B),A) ) ) ) ) )).

fof(t40_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => ( ( v1_comseq_1(A)
                  & v1_comseq_1(B) )
               => k8_comseq_1(C,A) = k8_comseq_1(k3_comseq_1(k5_numbers,C,B),k3_comseq_1(k5_numbers,A,B)) ) ) ) ) )).

fof(t41_comseq_1,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) )
         => ( v1_comseq_1(B)
           => ( A = k5_complex1
              | v1_comseq_1(k4_comseq_1(k5_numbers,B,A)) ) ) ) ) )).

fof(t42_comseq_1,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(k5_comseq_1(k5_numbers,A)) ) ) )).

fof(t43_comseq_1,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) )
         => ( v1_comseq_1(B)
           => ( A = k5_complex1
              | k7_comseq_1(k4_comseq_1(k5_numbers,B,A)) = k4_comseq_1(k5_numbers,k7_comseq_1(B),k12_complex1(A)) ) ) ) ) )).

fof(t44_comseq_1,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)
       => k7_comseq_1(k5_comseq_1(k5_numbers,A)) = k4_comseq_1(k5_numbers,k7_comseq_1(A),k10_complex1(k6_complex1)) ) ) )).

fof(t45_comseq_1,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_comseq_1(A)
           => ( k5_comseq_1(k5_numbers,k8_comseq_1(B,A)) = k8_comseq_1(k5_comseq_1(k5_numbers,B),A)
              & k8_comseq_1(B,k5_comseq_1(k5_numbers,A)) = k5_comseq_1(k5_numbers,k8_comseq_1(B,A)) ) ) ) ) )).

fof(t46_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => ( k2_comseq_1(k5_numbers,k8_comseq_1(A,B),k8_comseq_1(C,B)) = k8_comseq_1(k2_comseq_1(k5_numbers,A,C),B)
                & k6_comseq_1(k5_numbers,k8_comseq_1(A,B),k8_comseq_1(C,B)) = k8_comseq_1(k6_comseq_1(k5_numbers,A,C),B) ) ) ) ) )).

fof(t47_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k2_numbers)
                    & m2_relset_1(D,k5_numbers,k2_numbers) )
                 => ( ( v1_comseq_1(A)
                      & v1_comseq_1(B) )
                   => ( k2_comseq_1(k5_numbers,k8_comseq_1(C,A),k8_comseq_1(D,B)) = k8_comseq_1(k2_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,C,B),k3_comseq_1(k5_numbers,D,A)),k3_comseq_1(k5_numbers,A,B))
                      & k6_comseq_1(k5_numbers,k8_comseq_1(C,A),k8_comseq_1(D,B)) = k8_comseq_1(k6_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,C,B),k3_comseq_1(k5_numbers,D,A)),k3_comseq_1(k5_numbers,A,B)) ) ) ) ) ) ) )).

fof(t48_comseq_1,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) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k2_numbers)
                    & m2_relset_1(D,k5_numbers,k2_numbers) )
                 => ( ( v1_comseq_1(A)
                      & v1_comseq_1(B)
                      & v1_comseq_1(C) )
                   => k8_comseq_1(k8_comseq_1(D,A),k8_comseq_1(B,C)) = k8_comseq_1(k3_comseq_1(k5_numbers,D,C),k3_comseq_1(k5_numbers,A,B)) ) ) ) ) ) )).

fof(t49_comseq_1,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) )
         => k9_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B)) = k11_seq_1(k9_comseq_1(k5_numbers,A),k9_comseq_1(k5_numbers,B)) ) ) )).

fof(t50_comseq_1,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)
       => v2_relat_1(k9_comseq_1(k5_numbers,A)) ) ) )).

fof(t51_comseq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => k18_seq_1(k9_comseq_1(k5_numbers,A)) = k9_comseq_1(k5_numbers,k7_comseq_1(A)) ) )).

fof(t52_comseq_1,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) )
         => k9_comseq_1(k5_numbers,k8_comseq_1(A,B)) = k19_seq_1(k9_comseq_1(k5_numbers,A),k9_comseq_1(k5_numbers,B)) ) ) )).

fof(t53_comseq_1,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) )
         => k9_comseq_1(k5_numbers,k4_comseq_1(k5_numbers,B,A)) = k14_seq_1(k9_comseq_1(k5_numbers,B),k17_complex1(A)) ) ) )).

fof(s1_comseq_1,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)
         => k1_comseq_1(A,B) = f1_s1_comseq_1(B) ) ) )).

fof(dt_k1_comseq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m1_relset_1(A,k5_numbers,k2_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_subset_1(k1_comseq_1(A,B),k2_numbers) ) )).

fof(redefinition_k1_comseq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m1_relset_1(A,k5_numbers,k2_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k1_comseq_1(A,B) = k1_funct_1(A,B) ) )).

fof(dt_k2_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers)
        & v1_funct_1(C)
        & m1_relset_1(C,A,k2_numbers) )
     => ( v1_funct_1(k2_comseq_1(A,B,C))
        & m2_relset_1(k2_comseq_1(A,B,C),A,k2_numbers) ) ) )).

fof(commutativity_k2_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers)
        & v1_funct_1(C)
        & m1_relset_1(C,A,k2_numbers) )
     => k2_comseq_1(A,B,C) = k2_comseq_1(A,C,B) ) )).

fof(dt_k3_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers)
        & v1_funct_1(C)
        & m1_relset_1(C,A,k2_numbers) )
     => ( v1_funct_1(k3_comseq_1(A,B,C))
        & m2_relset_1(k3_comseq_1(A,B,C),A,k2_numbers) ) ) )).

fof(commutativity_k3_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers)
        & v1_funct_1(C)
        & m1_relset_1(C,A,k2_numbers) )
     => k3_comseq_1(A,B,C) = k3_comseq_1(A,C,B) ) )).

fof(dt_k4_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers)
        & v1_xcmplx_0(C) )
     => ( v1_funct_1(k4_comseq_1(A,B,C))
        & m2_relset_1(k4_comseq_1(A,B,C),A,k2_numbers) ) ) )).

fof(dt_k5_comseq_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers) )
     => ( v1_funct_1(k5_comseq_1(A,B))
        & m2_relset_1(k5_comseq_1(A,B),A,k2_numbers) ) ) )).

fof(dt_k6_comseq_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers)
        & v1_funct_1(C)
        & m1_relset_1(C,A,k2_numbers) )
     => ( v1_funct_1(k6_comseq_1(A,B,C))
        & m2_relset_1(k6_comseq_1(A,B,C),A,k2_numbers) ) ) )).

fof(dt_k7_comseq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m1_relset_1(A,k5_numbers,k2_numbers) )
     => ( v1_funct_1(k7_comseq_1(A))
        & v1_funct_2(k7_comseq_1(A),k5_numbers,k2_numbers)
        & m2_relset_1(k7_comseq_1(A),k5_numbers,k2_numbers) ) ) )).

fof(dt_k8_comseq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m1_relset_1(A,k5_numbers,k2_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k2_numbers)
        & m1_relset_1(B,k5_numbers,k2_numbers) )
     => ( v1_funct_1(k8_comseq_1(A,B))
        & v1_funct_2(k8_comseq_1(A,B),k5_numbers,k2_numbers)
        & m2_relset_1(k8_comseq_1(A,B),k5_numbers,k2_numbers) ) ) )).

fof(dt_k9_comseq_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k2_numbers) )
     => ( v1_funct_1(k9_comseq_1(A,B))
        & m2_relset_1(k9_comseq_1(A,B),A,k1_numbers) ) ) )).
%------------------------------------------------------------------------------