SET007 Axioms: SET007+81.ax


%------------------------------------------------------------------------------
% File     : SET007+81 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Real Sequences and Basic Operations on Them
% 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    : seq_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  130 (   8 unit)
%            Number of atoms       :  906 ( 109 equality)
%            Maximal formula depth :   12 (   7 average)
%            Number of connectives :  778 (   2 ~  ;   1  |; 514  &)
%                                         (  21 <=>; 240 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   18 (   1 propositional; 0-3 arity)
%            Number of functors    :   53 (  11 constant; 0-4 arity)
%            Number of variables   :  302 (   0 singleton; 296 !;   6 ?)
%            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(cc1_seq_1,axiom,(
    ! [A,B] :
      ( v2_membered(B)
     => ! [C] :
          ( m1_relset_1(C,A,B)
         => ( v1_funct_1(C)
           => ( v1_funct_1(C)
              & v1_seq_1(C) ) ) ) ) )).

fof(rc1_seq_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_seq_1(A) ) )).

fof(fc1_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_xcmplx_0(k1_funct_1(A,B))
        & v1_xreal_0(k1_funct_1(A,B)) ) ) )).

fof(fc2_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_seq_1(k7_relat_1(A,B)) ) ) )).

fof(fc3_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k3_seq_1(A,B))
        & v1_funct_1(k3_seq_1(A,B))
        & v1_seq_1(k3_seq_1(A,B)) ) ) )).

fof(fc4_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k4_seq_1(A,B))
        & v1_funct_1(k4_seq_1(A,B))
        & v1_seq_1(k4_seq_1(A,B)) ) ) )).

fof(fc5_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k5_seq_1(A,B))
        & v1_funct_1(k5_seq_1(A,B))
        & v1_seq_1(k5_seq_1(A,B)) ) ) )).

fof(fc6_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k12_seq_1(B,A))
        & v1_funct_1(k12_seq_1(B,A))
        & v1_seq_1(k12_seq_1(B,A)) ) ) )).

fof(fc7_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k15_seq_1(A))
        & v1_funct_1(k15_seq_1(A))
        & v1_seq_1(k15_seq_1(A)) ) ) )).

fof(fc8_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k20_seq_1(A))
        & v1_funct_1(k20_seq_1(A))
        & v1_seq_1(k20_seq_1(A)) ) ) )).

fof(t1_seq_1,axiom,(
    $true )).

fof(t2_seq_1,axiom,(
    $true )).

fof(t3_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & m2_relset_1(A,k5_numbers,k1_numbers) )
      <=> ( k1_relat_1(A) = k5_numbers
          & ! [B] :
              ( r2_hidden(B,k5_numbers)
             => v1_xreal_0(k1_funct_1(A,B)) ) ) ) ) )).

fof(t4_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & m2_relset_1(A,k5_numbers,k1_numbers) )
      <=> ( k1_relat_1(A) = k5_numbers
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => v1_xreal_0(k1_funct_1(A,B)) ) ) ) ) )).

fof(d1_seq_1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v1_seq_1(A)
      <=> r1_tarski(k2_relat_1(A),k1_numbers) ) ) )).

fof(d2_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_relat_1(A)
      <=> r1_tarski(k5_relset_1(k5_numbers,k1_numbers,A),k4_xboole_0(k1_numbers,k1_tarski(np__0))) ) ) )).

fof(t5_seq_1,axiom,(
    $true )).

fof(t6_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_relat_1(A)
      <=> ! [B] :
            ~ ( r2_hidden(B,k5_numbers)
              & k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 ) ) ) )).

fof(t7_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_relat_1(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) != np__0 ) ) ) )).

fof(t8_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ! [C] :
                ( r2_hidden(C,k5_numbers)
               => k2_seq_1(k5_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,B,C) )
           => A = B ) ) ) )).

fof(t9_seq_1,axiom,(
    $true )).

fof(t10_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,k1_numbers)
          & m2_relset_1(B,k5_numbers,k1_numbers)
          & k5_relset_1(k5_numbers,k1_numbers,B) = k1_tarski(A) ) ) )).

fof(d3_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_seq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( C = k3_seq_1(A,B)
              <=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
                  & ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                     => k1_funct_1(C,D) = k3_real_1(k1_seq_1(A,D),k1_seq_1(B,D)) ) ) ) ) ) ) )).

fof(d4_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_seq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( C = k4_seq_1(A,B)
              <=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
                  & ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                     => k1_funct_1(C,D) = k5_real_1(k1_seq_1(A,D),k1_seq_1(B,D)) ) ) ) ) ) ) )).

fof(d5_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_seq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( C = k5_seq_1(A,B)
              <=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
                  & ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                     => k1_funct_1(C,D) = k4_real_1(k1_seq_1(A,D),k1_seq_1(B,D)) ) ) ) ) ) ) )).

fof(t11_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( A = k6_seq_1(k5_numbers,k1_numbers,B,C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,A,D) = k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) )).

fof(t12_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( A = k8_seq_1(k5_numbers,k1_numbers,B,C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,A,D) = k4_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) )).

fof(d6_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( C = k12_seq_1(A,B)
              <=> ( k1_relat_1(C) = k1_relat_1(A)
                  & ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                     => k1_funct_1(C,D) = k3_xcmplx_0(B,k1_seq_1(A,D)) ) ) ) ) ) ) )).

fof(t13_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( B = k14_seq_1(C,A)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,B,D) = k3_xcmplx_0(A,k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) )).

fof(d7_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k15_seq_1(A)
          <=> ( k1_relat_1(B) = k1_relat_1(A)
              & ! [C] :
                  ( r2_hidden(C,k1_relat_1(B))
                 => k1_funct_1(B,C) = k1_real_1(k1_seq_1(A,C)) ) ) ) ) ) )).

fof(t14_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( A = k17_seq_1(B)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_seq_1(k5_numbers,k1_numbers,A,C) = k1_real_1(k2_seq_1(k5_numbers,k1_numbers,B,C)) ) ) ) ) )).

fof(t15_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k10_seq_1(A,B) = k9_seq_1(A,k17_seq_1(B)) ) ) )).

fof(d8_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( B = k18_seq_1(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_real_1(k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) )).

fof(d9_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k19_seq_1(A,B) = k11_seq_1(A,k18_seq_1(B)) ) ) )).

fof(d10_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k20_seq_1(A)
          <=> ( k1_relat_1(B) = k1_relat_1(A)
              & ! [C] :
                  ( r2_hidden(C,k1_relat_1(B))
                 => k1_funct_1(B,C) = k18_complex1(k1_seq_1(A,C)) ) ) ) ) ) )).

fof(t16_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( A = k22_seq_1(B)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_seq_1(k5_numbers,k1_numbers,A,C) = k18_complex1(k2_seq_1(k5_numbers,k1_numbers,B,C)) ) ) ) ) )).

fof(t17_seq_1,axiom,(
    $true )).

fof(t18_seq_1,axiom,(
    $true )).

fof(t19_seq_1,axiom,(
    $true )).

fof(t20_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k9_seq_1(k9_seq_1(A,B),C) = k9_seq_1(A,k9_seq_1(B,C)) ) ) ) )).

fof(t21_seq_1,axiom,(
    $true )).

fof(t22_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k11_seq_1(k11_seq_1(A,B),C) = k11_seq_1(A,k11_seq_1(B,C)) ) ) ) )).

fof(t23_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k11_seq_1(k9_seq_1(A,B),C) = k9_seq_1(k11_seq_1(A,C),k11_seq_1(B,C)) ) ) ) )).

fof(t24_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k11_seq_1(A,k9_seq_1(B,C)) = k9_seq_1(k11_seq_1(A,B),k11_seq_1(A,C)) ) ) ) )).

fof(t25_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k17_seq_1(A) = k14_seq_1(A,k1_real_1(np__1)) ) )).

fof(t26_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k14_seq_1(k11_seq_1(B,C),A) = k11_seq_1(k14_seq_1(B,A),C) ) ) ) )).

fof(t27_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k14_seq_1(k11_seq_1(B,C),A) = k11_seq_1(B,k14_seq_1(C,A)) ) ) ) )).

fof(t28_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k11_seq_1(k10_seq_1(A,B),C) = k10_seq_1(k11_seq_1(A,C),k11_seq_1(B,C)) ) ) ) )).

fof(t29_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k10_seq_1(k11_seq_1(A,B),k11_seq_1(A,C)) = k11_seq_1(A,k10_seq_1(B,C)) ) ) ) )).

fof(t30_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k14_seq_1(k9_seq_1(B,C),A) = k9_seq_1(k14_seq_1(B,A),k14_seq_1(C,A)) ) ) ) )).

fof(t31_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k14_seq_1(C,k3_xcmplx_0(A,B)) = k14_seq_1(k14_seq_1(C,B),A) ) ) ) )).

fof(t32_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k14_seq_1(k10_seq_1(B,C),A) = k10_seq_1(k14_seq_1(B,A),k14_seq_1(C,A)) ) ) ) )).

fof(t33_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k14_seq_1(k19_seq_1(B,C),A) = k19_seq_1(k14_seq_1(B,A),C) ) ) ) )).

fof(t34_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k10_seq_1(A,k9_seq_1(B,C)) = k10_seq_1(k10_seq_1(A,B),C) ) ) ) )).

fof(t35_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k14_seq_1(A,np__1) = A ) )).

fof(t36_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k17_seq_1(k17_seq_1(A)) = A ) )).

fof(t37_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k10_seq_1(A,k17_seq_1(B)) = k9_seq_1(A,B) ) ) )).

fof(t38_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k10_seq_1(A,k10_seq_1(B,C)) = k9_seq_1(k10_seq_1(A,B),C) ) ) ) )).

fof(t39_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k9_seq_1(A,k10_seq_1(B,C)) = k10_seq_1(k9_seq_1(A,B),C) ) ) ) )).

fof(t40_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( k11_seq_1(k17_seq_1(A),B) = k17_seq_1(k11_seq_1(A,B))
            & k11_seq_1(A,k17_seq_1(B)) = k17_seq_1(k11_seq_1(A,B)) ) ) ) )).

fof(t41_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_relat_1(A)
       => v2_relat_1(k18_seq_1(A)) ) ) )).

fof(t42_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k18_seq_1(k18_seq_1(A)) = A ) )).

fof(t43_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( v2_relat_1(A)
              & v2_relat_1(B) )
          <=> v2_relat_1(k11_seq_1(A,B)) ) ) ) )).

fof(t44_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k11_seq_1(k18_seq_1(A),k18_seq_1(B)) = k18_seq_1(k11_seq_1(A,B)) ) ) )).

fof(t45_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v2_relat_1(A)
           => k11_seq_1(k19_seq_1(B,A),A) = B ) ) ) )).

fof(t46_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => k11_seq_1(k19_seq_1(A,B),k19_seq_1(C,D)) = k19_seq_1(k11_seq_1(A,C),k11_seq_1(B,D)) ) ) ) ) )).

fof(t47_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( v2_relat_1(A)
              & v2_relat_1(B) )
           => v2_relat_1(k19_seq_1(A,B)) ) ) ) )).

fof(t48_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k18_seq_1(k19_seq_1(A,B)) = k19_seq_1(B,A) ) ) )).

fof(t49_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k11_seq_1(A,k19_seq_1(B,C)) = k19_seq_1(k11_seq_1(A,B),C) ) ) ) )).

fof(t50_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k19_seq_1(A,k19_seq_1(B,C)) = k19_seq_1(k11_seq_1(A,C),B) ) ) ) )).

fof(t51_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( v2_relat_1(A)
               => k19_seq_1(B,C) = k19_seq_1(k11_seq_1(B,A),k11_seq_1(C,A)) ) ) ) ) )).

fof(t52_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v2_relat_1(B)
           => ( A = np__0
              | v2_relat_1(k14_seq_1(B,A)) ) ) ) ) )).

fof(t53_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_relat_1(A)
       => v2_relat_1(k17_seq_1(A)) ) ) )).

fof(t54_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k18_seq_1(k14_seq_1(B,A)) = k14_seq_1(k18_seq_1(B),k5_xcmplx_0(A)) ) ) )).

fof(t55_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k18_seq_1(k17_seq_1(A)) = k14_seq_1(k18_seq_1(A),k1_real_1(np__1)) ) )).

fof(t56_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( k17_seq_1(k19_seq_1(A,B)) = k19_seq_1(k17_seq_1(A),B)
            & k19_seq_1(A,k17_seq_1(B)) = k17_seq_1(k19_seq_1(A,B)) ) ) ) )).

fof(t57_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( k9_seq_1(k19_seq_1(A,B),k19_seq_1(C,B)) = k19_seq_1(k9_seq_1(A,C),B)
                & k10_seq_1(k19_seq_1(A,B),k19_seq_1(C,B)) = k19_seq_1(k10_seq_1(A,C),B) ) ) ) ) )).

fof(t58_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => ( ( v2_relat_1(A)
                      & v2_relat_1(B) )
                   => ( k9_seq_1(k19_seq_1(C,A),k19_seq_1(D,B)) = k19_seq_1(k9_seq_1(k11_seq_1(C,B),k11_seq_1(D,A)),k11_seq_1(A,B))
                      & k10_seq_1(k19_seq_1(C,A),k19_seq_1(D,B)) = k19_seq_1(k10_seq_1(k11_seq_1(C,B),k11_seq_1(D,A)),k11_seq_1(A,B)) ) ) ) ) ) ) )).

fof(t59_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => k19_seq_1(k19_seq_1(A,B),k19_seq_1(C,D)) = k19_seq_1(k11_seq_1(A,D),k11_seq_1(B,C)) ) ) ) ) )).

fof(t60_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k22_seq_1(k11_seq_1(A,B)) = k11_seq_1(k22_seq_1(A),k22_seq_1(B)) ) ) )).

fof(t61_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_relat_1(A)
       => v2_relat_1(k22_seq_1(A)) ) ) )).

fof(t62_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k18_seq_1(k22_seq_1(A)) = k22_seq_1(k18_seq_1(A)) ) )).

fof(t63_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k22_seq_1(k19_seq_1(A,B)) = k19_seq_1(k22_seq_1(A),k22_seq_1(B)) ) ) )).

fof(t64_seq_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k22_seq_1(k14_seq_1(B,A)) = k14_seq_1(k22_seq_1(B),k18_complex1(A)) ) ) )).

fof(s1_seq_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_numbers)
      & m2_relset_1(A,k5_numbers,k1_numbers)
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_seq_1(k5_numbers,k1_numbers,A,B) = f1_s1_seq_1(B) ) ) )).

fof(s2_seq_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & m2_relset_1(A,f1_s2_seq_1,f2_s2_seq_1)
      & ! [B] :
          ( m1_subset_1(B,f1_s2_seq_1)
         => ( r2_hidden(B,k4_relset_1(f1_s2_seq_1,f2_s2_seq_1,A))
          <=> ? [C] :
                ( m1_subset_1(C,f2_s2_seq_1)
                & p1_s2_seq_1(B,C) ) ) )
      & ! [B] :
          ( m1_subset_1(B,f1_s2_seq_1)
         => ( r2_hidden(B,k4_relset_1(f1_s2_seq_1,f2_s2_seq_1,A))
           => p1_s2_seq_1(B,k1_funct_1(A,B)) ) ) ) )).

fof(s3_seq_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & m2_relset_1(A,f1_s3_seq_1,f2_s3_seq_1)
      & ! [B] :
          ( m1_subset_1(B,f1_s3_seq_1)
         => ( r2_hidden(B,k4_relset_1(f1_s3_seq_1,f2_s3_seq_1,A))
          <=> p1_s3_seq_1(B) ) )
      & ! [B] :
          ( m1_subset_1(B,f1_s3_seq_1)
         => ( r2_hidden(B,k4_relset_1(f1_s3_seq_1,f2_s3_seq_1,A))
           => k1_funct_1(A,B) = f3_s3_seq_1(B) ) ) ) )).

fof(s4_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,f1_s4_seq_1,f2_s4_seq_1) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,f1_s4_seq_1,f2_s4_seq_1) )
         => ( ( k4_relset_1(f1_s4_seq_1,f2_s4_seq_1,A) = f3_s4_seq_1
              & ! [C] :
                  ( m1_subset_1(C,f1_s4_seq_1)
                 => ( r2_hidden(C,k4_relset_1(f1_s4_seq_1,f2_s4_seq_1,A))
                   => k1_funct_1(A,C) = f4_s4_seq_1(C) ) )
              & k4_relset_1(f1_s4_seq_1,f2_s4_seq_1,B) = f3_s4_seq_1
              & ! [C] :
                  ( m1_subset_1(C,f1_s4_seq_1)
                 => ( r2_hidden(C,k4_relset_1(f1_s4_seq_1,f2_s4_seq_1,B))
                   => k1_funct_1(B,C) = f4_s4_seq_1(C) ) ) )
           => A = B ) ) ) )).

fof(dt_k1_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => m1_subset_1(k1_seq_1(A,B),k1_numbers) ) )).

fof(redefinition_k1_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => k1_seq_1(A,B) = k1_funct_1(A,B) ) )).

fof(dt_k2_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) )).

fof(redefinition_k2_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k3_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k3_seq_1(A,B))
        & v1_funct_1(k3_seq_1(A,B)) ) ) )).

fof(commutativity_k3_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => k3_seq_1(A,B) = k3_seq_1(B,A) ) )).

fof(dt_k4_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k4_seq_1(A,B))
        & v1_funct_1(k4_seq_1(A,B)) ) ) )).

fof(dt_k5_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => ( v1_relat_1(k5_seq_1(A,B))
        & v1_funct_1(k5_seq_1(A,B)) ) ) )).

fof(commutativity_k5_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_seq_1(B) )
     => k5_seq_1(A,B) = k5_seq_1(B,A) ) )).

fof(dt_k6_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => ( v1_funct_1(k6_seq_1(A,B,C,D))
        & m2_relset_1(k6_seq_1(A,B,C,D),A,k1_numbers) ) ) )).

fof(commutativity_k6_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => k6_seq_1(A,B,C,D) = k6_seq_1(A,B,D,C) ) )).

fof(redefinition_k6_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => k6_seq_1(A,B,C,D) = k3_seq_1(C,D) ) )).

fof(dt_k7_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => ( v1_funct_1(k7_seq_1(A,B,C,D))
        & m2_relset_1(k7_seq_1(A,B,C,D),A,k1_numbers) ) ) )).

fof(redefinition_k7_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => k7_seq_1(A,B,C,D) = k4_seq_1(C,D) ) )).

fof(dt_k8_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => ( v1_funct_1(k8_seq_1(A,B,C,D))
        & m2_relset_1(k8_seq_1(A,B,C,D),A,k1_numbers) ) ) )).

fof(commutativity_k8_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => k8_seq_1(A,B,C,D) = k8_seq_1(A,B,D,C) ) )).

fof(redefinition_k8_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => k8_seq_1(A,B,C,D) = k5_seq_1(C,D) ) )).

fof(dt_k9_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k9_seq_1(A,B))
        & v1_funct_2(k9_seq_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k9_seq_1(A,B),k5_numbers,k1_numbers) ) ) )).

fof(commutativity_k9_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => k9_seq_1(A,B) = k9_seq_1(B,A) ) )).

fof(redefinition_k9_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => k9_seq_1(A,B) = k3_seq_1(A,B) ) )).

fof(dt_k10_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k10_seq_1(A,B))
        & v1_funct_2(k10_seq_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k10_seq_1(A,B),k5_numbers,k1_numbers) ) ) )).

fof(redefinition_k10_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => k10_seq_1(A,B) = k4_seq_1(A,B) ) )).

fof(dt_k11_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k11_seq_1(A,B))
        & v1_funct_2(k11_seq_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k11_seq_1(A,B),k5_numbers,k1_numbers) ) ) )).

fof(commutativity_k11_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => k11_seq_1(A,B) = k11_seq_1(B,A) ) )).

fof(redefinition_k11_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => k11_seq_1(A,B) = k5_seq_1(A,B) ) )).

fof(dt_k12_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_xreal_0(B) )
     => ( v1_relat_1(k12_seq_1(A,B))
        & v1_funct_1(k12_seq_1(A,B)) ) ) )).

fof(dt_k13_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_xreal_0(D) )
     => ( v1_funct_1(k13_seq_1(A,B,C,D))
        & m2_relset_1(k13_seq_1(A,B,C,D),A,k1_numbers) ) ) )).

fof(redefinition_k13_seq_1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B)
        & v1_xreal_0(D) )
     => k13_seq_1(A,B,C,D) = k12_seq_1(C,D) ) )).

fof(dt_k14_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_xreal_0(B) )
     => ( v1_funct_1(k14_seq_1(A,B))
        & v1_funct_2(k14_seq_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k14_seq_1(A,B),k5_numbers,k1_numbers) ) ) )).

fof(redefinition_k14_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_xreal_0(B) )
     => k14_seq_1(A,B) = k12_seq_1(A,B) ) )).

fof(dt_k15_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k15_seq_1(A))
        & v1_funct_1(k15_seq_1(A)) ) ) )).

fof(dt_k16_seq_1,axiom,(
    ! [A,B,C] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => ( v1_funct_1(k16_seq_1(A,B,C))
        & m2_relset_1(k16_seq_1(A,B,C),A,k1_numbers) ) ) )).

fof(redefinition_k16_seq_1,axiom,(
    ! [A,B,C] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => k16_seq_1(A,B,C) = k15_seq_1(C) ) )).

fof(dt_k17_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k17_seq_1(A))
        & v1_funct_2(k17_seq_1(A),k5_numbers,k1_numbers)
        & m2_relset_1(k17_seq_1(A),k5_numbers,k1_numbers) ) ) )).

fof(redefinition_k17_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => k17_seq_1(A) = k15_seq_1(A) ) )).

fof(dt_k18_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k18_seq_1(A))
        & v1_funct_2(k18_seq_1(A),k5_numbers,k1_numbers)
        & m2_relset_1(k18_seq_1(A),k5_numbers,k1_numbers) ) ) )).

fof(dt_k19_seq_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k19_seq_1(A,B))
        & v1_funct_2(k19_seq_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k19_seq_1(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k20_seq_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k20_seq_1(A))
        & v1_funct_1(k20_seq_1(A)) ) ) )).

fof(dt_k21_seq_1,axiom,(
    ! [A,B,C] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => ( v1_funct_1(k21_seq_1(A,B,C))
        & m2_relset_1(k21_seq_1(A,B,C),A,k1_numbers) ) ) )).

fof(redefinition_k21_seq_1,axiom,(
    ! [A,B,C] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => k21_seq_1(A,B,C) = k20_seq_1(C) ) )).

fof(dt_k22_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k22_seq_1(A))
        & v1_funct_2(k22_seq_1(A),k5_numbers,k1_numbers)
        & m2_relset_1(k22_seq_1(A),k5_numbers,k1_numbers) ) ) )).

fof(redefinition_k22_seq_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => k22_seq_1(A) = k20_seq_1(A) ) )).
%------------------------------------------------------------------------------