SET007 Axioms: SET007+86.ax


%------------------------------------------------------------------------------
% File     : SET007+86 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Convergent Real Sequences
% Version  : [Urb08] axioms.
% English  : Upper and Lower Bound of Sets of Real Numbers

% 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_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   79 (  13 unit)
%            Number of atoms       :  529 (  50 equality)
%            Maximal formula depth :   20 (   7 average)
%            Number of connectives :  522 (  72 ~  ;  18  |; 198  &)
%                                         (   7 <=>; 227 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-3 arity)
%            Number of functors    :   32 (   4 constant; 0-4 arity)
%            Number of variables   :  178 (   0 singleton; 165 !;  13 ?)
%            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_4,axiom,(
    ! [A] :
      ( m1_relset_1(A,k5_numbers,k1_numbers)
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v5_seqm_3(A) )
       => ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v1_seq_1(A)
          & v4_seq_2(A) ) ) ) )).

fof(t1_seq_4,axiom,(
    $true )).

fof(t2_seq_4,axiom,(
    $true )).

fof(t3_seq_4,axiom,(
    $true )).

fof(t4_seq_4,axiom,(
    $true )).

fof(t5_seq_4,axiom,(
    $true )).

fof(t6_seq_4,axiom,(
    $true )).

fof(t7_seq_4,axiom,(
    $true )).

fof(t8_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ~ ( ! [C] :
                  ( v1_xreal_0(C)
                 => ! [D] :
                      ( v1_xreal_0(D)
                     => ~ ( r2_hidden(C,A)
                          & r2_hidden(D,B)
                          & r1_xreal_0(D,C) ) ) )
              & ! [C] :
                  ( v1_xreal_0(C)
                 => ? [D] :
                      ( v1_xreal_0(D)
                      & ? [E] :
                          ( v1_xreal_0(E)
                          & r2_hidden(D,A)
                          & r2_hidden(E,B)
                          & ~ ( r1_xreal_0(D,C)
                              & r1_xreal_0(C,E) ) ) ) ) ) ) ) )).

fof(t9_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( ! [C] :
                ( v1_xreal_0(C)
               => ( r2_hidden(C,B)
                 => r2_hidden(k2_xcmplx_0(C,A),B) ) )
           => ( r1_xreal_0(A,np__0)
              | ! [C] :
                  ( v1_xreal_0(C)
                 => ~ r2_hidden(C,B) )
              | ! [C] :
                  ( v1_xreal_0(C)
                 => ? [D] :
                      ( v1_xreal_0(D)
                      & r2_hidden(D,B)
                      & ~ r1_xreal_0(D,C) ) ) ) ) ) ) )).

fof(t10_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r1_xreal_0(B,A) ) ) )).

fof(d1_seq_4,axiom,(
    ! [A] :
      ( v2_membered(A)
     => ( v1_seq_4(A)
      <=> ? [B] :
            ( v1_xreal_0(B)
            & ! [C] :
                ( v1_xreal_0(C)
               => ( r2_hidden(C,A)
                 => r1_xreal_0(C,B) ) ) ) ) ) )).

fof(d2_seq_4,axiom,(
    ! [A] :
      ( v2_membered(A)
     => ( v2_seq_4(A)
      <=> ? [B] :
            ( v1_xreal_0(B)
            & ! [C] :
                ( v1_xreal_0(C)
               => ( r2_hidden(C,A)
                 => r1_xreal_0(B,C) ) ) ) ) ) )).

fof(d3_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v3_seq_4(A)
      <=> ( v2_seq_4(A)
          & v1_seq_4(A) ) ) ) )).

fof(t11_seq_4,axiom,(
    $true )).

fof(t12_seq_4,axiom,(
    $true )).

fof(t13_seq_4,axiom,(
    $true )).

fof(t14_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v3_seq_4(A)
      <=> ? [B] :
            ( v1_xreal_0(B)
            & ~ r1_xreal_0(B,np__0)
            & ! [C] :
                ( v1_xreal_0(C)
               => ~ ( r2_hidden(C,A)
                    & r1_xreal_0(B,k18_complex1(C)) ) ) ) ) ) )).

fof(t15_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => v3_seq_4(k1_seq_4(A)) ) )).

fof(t16_seq_4,axiom,(
    ! [A] :
      ( v2_membered(A)
     => ~ ( ~ v1_xboole_0(A)
          & v1_seq_4(A)
          & ! [B] :
              ( v1_xreal_0(B)
             => ~ ( ! [C] :
                      ( v1_xreal_0(C)
                     => ( r2_hidden(C,A)
                       => r1_xreal_0(C,B) ) )
                  & ! [C] :
                      ( v1_xreal_0(C)
                     => ~ ( ~ r1_xreal_0(C,np__0)
                          & ! [D] :
                              ( v1_xreal_0(D)
                             => ~ ( r2_hidden(D,A)
                                  & ~ r1_xreal_0(D,k6_xcmplx_0(B,C)) ) ) ) ) ) ) ) ) )).

fof(t17_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v2_membered(C)
             => ( ( ! [D] :
                      ( v1_xreal_0(D)
                     => ( r2_hidden(D,C)
                       => r1_xreal_0(D,A) ) )
                  & ! [D] :
                      ( v1_xreal_0(D)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( v1_xreal_0(E)
                             => ~ ( r2_hidden(E,C)
                                  & ~ r1_xreal_0(E,k6_xcmplx_0(A,D)) ) ) ) )
                  & ! [D] :
                      ( v1_xreal_0(D)
                     => ( r2_hidden(D,C)
                       => r1_xreal_0(D,B) ) )
                  & ! [D] :
                      ( v1_xreal_0(D)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( v1_xreal_0(E)
                             => ~ ( r2_hidden(E,C)
                                  & ~ r1_xreal_0(E,k6_xcmplx_0(B,D)) ) ) ) ) )
               => A = B ) ) ) ) )).

fof(t18_seq_4,axiom,(
    ! [A] :
      ( v2_membered(A)
     => ~ ( ~ v1_xboole_0(A)
          & v2_seq_4(A)
          & ! [B] :
              ( v1_xreal_0(B)
             => ~ ( ! [C] :
                      ( v1_xreal_0(C)
                     => ( r2_hidden(C,A)
                       => r1_xreal_0(B,C) ) )
                  & ! [C] :
                      ( v1_xreal_0(C)
                     => ~ ( ~ r1_xreal_0(C,np__0)
                          & ! [D] :
                              ( v1_xreal_0(D)
                             => ~ ( r2_hidden(D,A)
                                  & ~ r1_xreal_0(k2_xcmplx_0(B,C),D) ) ) ) ) ) ) ) ) )).

fof(t19_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v2_membered(C)
             => ( ( ! [D] :
                      ( v1_xreal_0(D)
                     => ( r2_hidden(D,C)
                       => r1_xreal_0(A,D) ) )
                  & ! [D] :
                      ( v1_xreal_0(D)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( v1_xreal_0(E)
                             => ~ ( r2_hidden(E,C)
                                  & ~ r1_xreal_0(k2_xcmplx_0(A,D),E) ) ) ) )
                  & ! [D] :
                      ( v1_xreal_0(D)
                     => ( r2_hidden(D,C)
                       => r1_xreal_0(B,D) ) )
                  & ! [D] :
                      ( v1_xreal_0(D)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( v1_xreal_0(E)
                             => ~ ( r2_hidden(E,C)
                                  & ~ r1_xreal_0(k2_xcmplx_0(B,D),E) ) ) ) ) )
               => A = B ) ) ) ) )).

fof(d4_seq_4,axiom,(
    ! [A] :
      ( v2_membered(A)
     => ( v1_seq_4(A)
       => ( v1_xboole_0(A)
          | ! [B] :
              ( v1_xreal_0(B)
             => ( B = k2_seq_4(A)
              <=> ( ! [C] :
                      ( v1_xreal_0(C)
                     => ( r2_hidden(C,A)
                       => r1_xreal_0(C,B) ) )
                  & ! [C] :
                      ( v1_xreal_0(C)
                     => ~ ( ~ r1_xreal_0(C,np__0)
                          & ! [D] :
                              ( v1_xreal_0(D)
                             => ~ ( r2_hidden(D,A)
                                  & ~ r1_xreal_0(D,k6_xcmplx_0(B,C)) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_seq_4,axiom,(
    ! [A] :
      ( v2_membered(A)
     => ( v2_seq_4(A)
       => ( v1_xboole_0(A)
          | ! [B] :
              ( v1_xreal_0(B)
             => ( B = k3_seq_4(A)
              <=> ( ! [C] :
                      ( v1_xreal_0(C)
                     => ( r2_hidden(C,A)
                       => r1_xreal_0(B,C) ) )
                  & ! [C] :
                      ( v1_xreal_0(C)
                     => ~ ( ~ r1_xreal_0(C,np__0)
                          & ! [D] :
                              ( v1_xreal_0(D)
                             => ~ ( r2_hidden(D,A)
                                  & ~ r1_xreal_0(k2_xcmplx_0(B,C),D) ) ) ) ) ) ) ) ) ) ) )).

fof(t20_seq_4,axiom,(
    $true )).

fof(t21_seq_4,axiom,(
    $true )).

fof(t22_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k5_seq_4(k1_seq_4(A)) = A
        & k4_seq_4(k1_seq_4(A)) = A ) ) )).

fof(t23_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k5_seq_4(k1_seq_4(A)) = k4_seq_4(k1_seq_4(A)) ) )).

fof(t24_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v3_seq_4(A)
       => ( v1_xboole_0(A)
          | r1_xreal_0(k5_seq_4(A),k4_seq_4(A)) ) ) ) )).

fof(t25_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v3_seq_4(A)
       => ( v1_xboole_0(A)
          | ( ~ ( ? [B] :
                    ( v1_xreal_0(B)
                    & ? [C] :
                        ( v1_xreal_0(C)
                        & r2_hidden(B,A)
                        & r2_hidden(C,A)
                        & C != B ) )
                & r1_xreal_0(k4_seq_4(A),k5_seq_4(A)) )
            & ~ ( ~ r1_xreal_0(k4_seq_4(A),k5_seq_4(A))
                & ! [B] :
                    ( v1_xreal_0(B)
                   => ! [C] :
                        ( v1_xreal_0(C)
                       => ~ ( r2_hidden(B,A)
                            & r2_hidden(C,A)
                            & C != B ) ) ) ) ) ) ) ) )).

fof(t26_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
       => v4_seq_2(k22_seq_1(A)) ) ) )).

fof(t27_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
       => k2_seq_2(k22_seq_1(A)) = k18_complex1(k2_seq_2(A)) ) ) )).

fof(t28_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v4_seq_2(k22_seq_1(A))
          & k2_seq_2(k22_seq_1(A)) = np__0 )
       => ( v4_seq_2(A)
          & k2_seq_2(A) = np__0 ) ) ) )).

fof(t29_seq_4,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) )
         => ( ( m1_seqm_3(A,B)
              & v4_seq_2(B) )
           => v4_seq_2(A) ) ) ) )).

fof(t30_seq_4,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) )
         => ( ( m1_seqm_3(A,B)
              & v4_seq_2(B) )
           => k2_seq_2(A) = k2_seq_2(B) ) ) ) )).

fof(t31_seq_4,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) )
         => ( v4_seq_2(A)
           => ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(C,D)
                      & k8_funct_2(k5_numbers,k1_numbers,B,D) != k8_funct_2(k5_numbers,k1_numbers,A,D) ) )
              | v4_seq_2(B) ) ) ) ) )).

fof(t32_seq_4,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) )
         => ( v4_seq_2(A)
           => ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(C,D)
                      & k8_funct_2(k5_numbers,k1_numbers,B,D) != k8_funct_2(k5_numbers,k1_numbers,A,D) ) )
              | k2_seq_2(A) = k2_seq_2(B) ) ) ) ) )).

fof(t33_seq_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v4_seq_2(B)
           => ( v4_seq_2(k1_seqm_3(B,A))
              & k2_seq_2(k1_seqm_3(B,A)) = k2_seq_2(B) ) ) ) ) )).

fof(t34_seq_4,axiom,(
    $true )).

fof(t35_seq_4,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) )
         => ( v4_seq_2(A)
           => ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => A != k1_seqm_3(B,C) )
              | v4_seq_2(B) ) ) ) ) )).

fof(t36_seq_4,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) )
         => ( v4_seq_2(A)
           => ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => A != k1_seqm_3(B,C) )
              | k2_seq_2(B) = k2_seq_2(A) ) ) ) ) )).

fof(t37_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ~ ( v4_seq_2(A)
          & k2_seq_2(A) != np__0
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ v2_relat_1(k1_seqm_3(A,B)) ) ) ) )).

fof(t38_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ~ ( v4_seq_2(A)
          & k2_seq_2(A) != np__0
          & ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,k5_numbers,k1_numbers)
                & m2_relset_1(B,k5_numbers,k1_numbers) )
             => ~ ( m1_seqm_3(B,A)
                  & v2_relat_1(B) ) ) ) ) )).

fof(t39_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v5_seqm_3(A)
       => v4_seq_2(A) ) ) )).

fof(t40_seq_4,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) )
         => ( ( ( v5_seqm_3(B)
                & r2_hidden(A,k2_relat_1(B)) )
              | ( v5_seqm_3(B)
                & ? [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                    & k8_funct_2(k5_numbers,k1_numbers,B,C) = A ) ) )
           => k2_seq_2(B) = A ) ) ) )).

fof(t41_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v5_seqm_3(A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_2(A) = k8_funct_2(k5_numbers,k1_numbers,A,B) ) ) ) )).

fof(t42_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
       => ( k2_seq_2(A) = np__0
          | ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,k5_numbers,k1_numbers)
                & m2_relset_1(B,k5_numbers,k1_numbers) )
             => ( ( m1_seqm_3(B,A)
                  & v2_relat_1(B) )
               => k2_seq_2(k18_seq_1(B)) = k2_real_1(k2_seq_2(A)) ) ) ) ) ) )).

fof(t43_seq_4,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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_xcmplx_0(C,A)) )
           => ( r1_xreal_0(A,np__0)
              | v4_seq_2(B) ) ) ) ) )).

fof(t44_seq_4,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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_xcmplx_0(C,A)) )
           => ( r1_xreal_0(A,np__0)
              | k2_seq_2(B) = np__0 ) ) ) ) )).

fof(t45_seq_4,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)
           => k8_funct_2(k5_numbers,k1_numbers,A,B) = k6_real_1(np__1,k1_nat_1(B,np__1)) )
       => ( v4_seq_2(A)
          & k2_seq_2(A) = np__0 ) ) ) )).

fof(t46_seq_4,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) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,k1_numbers,C,D) = k7_xcmplx_0(B,k2_xcmplx_0(D,A)) )
               => ( r1_xreal_0(A,np__0)
                  | ( v4_seq_2(C)
                    & k2_seq_2(C) = np__0 ) ) ) ) ) ) )).

fof(t47_seq_4,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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_xcmplx_0(k2_nat_1(C,C),A)) )
           => ( r1_xreal_0(A,np__0)
              | v4_seq_2(B) ) ) ) ) )).

fof(t48_seq_4,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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_xcmplx_0(k2_nat_1(C,C),A)) )
           => ( r1_xreal_0(A,np__0)
              | k2_seq_2(B) = np__0 ) ) ) ) )).

fof(t49_seq_4,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)
           => k8_funct_2(k5_numbers,k1_numbers,A,B) = k6_real_1(np__1,k1_nat_1(k2_nat_1(B,B),np__1)) )
       => ( v4_seq_2(A)
          & k2_seq_2(A) = np__0 ) ) ) )).

fof(t50_seq_4,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) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,k1_numbers,C,D) = k7_xcmplx_0(B,k2_xcmplx_0(k2_nat_1(D,D),A)) )
               => ( r1_xreal_0(A,np__0)
                  | ( v4_seq_2(C)
                    & k2_seq_2(C) = np__0 ) ) ) ) ) ) )).

fof(t51_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v3_seqm_3(A)
          & v1_seq_2(A) )
       => v4_seq_2(A) ) ) )).

fof(t52_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v4_seqm_3(A)
          & v2_seq_2(A) )
       => v4_seq_2(A) ) ) )).

fof(t53_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v6_seqm_3(A)
          & v3_seq_2(A) )
       => v4_seq_2(A) ) ) )).

fof(t54_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v1_seq_2(A)
          & v3_seqm_3(A) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,B),k2_seq_2(A)) ) ) ) )).

fof(t55_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v2_seq_2(A)
          & v4_seqm_3(A) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r1_xreal_0(k2_seq_2(A),k8_funct_2(k5_numbers,k1_numbers,A,B)) ) ) ) )).

fof(t56_seq_4,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)
          & v1_seqm_3(B)
          & v7_seqm_3(B)
          & m2_relset_1(B,k5_numbers,k1_numbers)
          & v6_seqm_3(k3_seqm_3(B,A)) ) ) )).

fof(t57_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ~ ( v3_seq_2(A)
          & ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,k5_numbers,k1_numbers)
                & m2_relset_1(B,k5_numbers,k1_numbers) )
             => ~ ( m1_seqm_3(B,A)
                  & v4_seq_2(B) ) ) ) ) )).

fof(t58_seq_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
      <=> ! [B] :
            ( v1_xreal_0(B)
           => ~ ( ~ r1_xreal_0(B,np__0)
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ? [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                        & r1_xreal_0(C,D)
                        & r1_xreal_0(B,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,A,D),k8_funct_2(k5_numbers,k1_numbers,A,C)))) ) ) ) ) ) ) )).

fof(t59_seq_4,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) )
         => ( ( v5_seqm_3(A)
              & v4_seq_2(B) )
           => ( k2_seq_2(k9_seq_1(A,B)) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,A,np__0),k2_seq_2(B))
              & k2_seq_2(k10_seq_1(A,B)) = k5_real_1(k8_funct_2(k5_numbers,k1_numbers,A,np__0),k2_seq_2(B))
              & k2_seq_2(k10_seq_1(B,A)) = k5_real_1(k2_seq_2(B),k8_funct_2(k5_numbers,k1_numbers,A,np__0))
              & k2_seq_2(k11_seq_1(A,B)) = k4_real_1(k8_funct_2(k5_numbers,k1_numbers,A,np__0),k2_seq_2(B)) ) ) ) ) )).

fof(t60_seq_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v2_membered(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ! [C] :
                ( v1_xreal_0(C)
               => ( r2_hidden(C,A)
                 => r1_xreal_0(B,C) ) )
           => r1_xreal_0(B,k3_seq_4(A)) ) ) ) )).

fof(t61_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v2_membered(B) )
         => ( ( ! [C] :
                  ( v1_xreal_0(C)
                 => ( r2_hidden(C,B)
                   => r1_xreal_0(A,C) ) )
              & ! [C] :
                  ( v1_xreal_0(C)
                 => ( ! [D] :
                        ( v1_xreal_0(D)
                       => ( r2_hidden(D,B)
                         => r1_xreal_0(C,D) ) )
                   => r1_xreal_0(C,A) ) ) )
           => A = k3_seq_4(B) ) ) ) )).

fof(t62_seq_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v2_membered(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ! [D] :
                    ( v1_xreal_0(D)
                   => ( r2_hidden(D,A)
                     => r1_xreal_0(D,C) ) )
               => r1_xreal_0(k2_seq_4(A),C) ) ) ) ) )).

fof(t63_seq_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v2_membered(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( ! [C] :
                  ( v1_xreal_0(C)
                 => ( r2_hidden(C,A)
                   => r1_xreal_0(C,B) ) )
              & ! [C] :
                  ( v1_xreal_0(C)
                 => ( ! [D] :
                        ( v1_xreal_0(D)
                       => ( r2_hidden(D,A)
                         => r1_xreal_0(D,C) ) )
                   => r1_xreal_0(B,C) ) ) )
           => B = k2_seq_4(A) ) ) ) )).

fof(t64_seq_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v2_membered(A) )
     => ! [B] :
          ( v2_membered(B)
         => ( ( r1_tarski(A,B)
              & v2_seq_4(B) )
           => r1_xreal_0(k3_seq_4(B),k3_seq_4(A)) ) ) ) )).

fof(t65_seq_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v2_membered(A) )
     => ! [B] :
          ( v2_membered(B)
         => ( ( r1_tarski(A,B)
              & v1_seq_4(B) )
           => r1_xreal_0(k2_seq_4(A),k2_seq_4(B)) ) ) ) )).

fof(dt_k1_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => m1_subset_1(k1_seq_4(A),k1_zfmisc_1(k1_numbers)) ) )).

fof(redefinition_k1_seq_4,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_seq_4(A) = k1_tarski(A) ) )).

fof(dt_k2_seq_4,axiom,(
    ! [A] :
      ( v2_membered(A)
     => v1_xreal_0(k2_seq_4(A)) ) )).

fof(dt_k3_seq_4,axiom,(
    ! [A] :
      ( v2_membered(A)
     => v1_xreal_0(k3_seq_4(A)) ) )).

fof(dt_k4_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => m1_subset_1(k4_seq_4(A),k1_numbers) ) )).

fof(redefinition_k4_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => k4_seq_4(A) = k2_seq_4(A) ) )).

fof(dt_k5_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => m1_subset_1(k5_seq_4(A),k1_numbers) ) )).

fof(redefinition_k5_seq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => k5_seq_4(A) = k3_seq_4(A) ) )).
%------------------------------------------------------------------------------