SET007 Axioms: SET007+145.ax


%------------------------------------------------------------------------------
% File     : SET007+145 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Series
% 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    : series_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   56 (   6 unit)
%            Number of atoms       :  417 (  43 equality)
%            Maximal formula depth :   13 (   7 average)
%            Number of connectives :  382 (  21 ~  ;  11  |; 188  &)
%                                         (   6 <=>; 156 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-3 arity)
%            Number of functors    :   28 (   5 constant; 0-4 arity)
%            Number of variables   :  129 (   0 singleton; 121 !;   8 ?)
%            Maximal term depth    :    6 (   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(t1_series_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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,C) = k3_power(A,k1_nat_1(C,np__1)) )
           => ( r1_xreal_0(A,np__0)
              | r1_xreal_0(np__1,A)
              | ( v4_seq_2(B)
                & k2_seq_2(B) = np__0 ) ) ) ) ) )).

fof(t2_series_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( B != np__0
           => k4_power(k18_complex1(B),A) = k18_complex1(k3_power(B,A)) ) ) ) )).

fof(t3_series_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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,C) = k3_power(A,k1_nat_1(C,np__1)) )
           => ( r1_xreal_0(np__1,k18_complex1(A))
              | ( v4_seq_2(B)
                & k2_seq_2(B) = np__0 ) ) ) ) ) )).

fof(d1_series_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 = k1_series_1(A)
          <=> ( k8_funct_2(k5_numbers,k1_numbers,B,np__0) = k8_funct_2(k5_numbers,k1_numbers,A,np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k8_funct_2(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,B,C),k8_funct_2(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) ) ) ) ) )).

fof(d2_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_series_1(A)
      <=> v4_seq_2(k1_series_1(A)) ) ) )).

fof(d3_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k2_series_1(A) = k2_seq_2(k1_series_1(A)) ) )).

fof(t4_series_1,axiom,(
    $true )).

fof(t5_series_1,axiom,(
    $true )).

fof(t6_series_1,axiom,(
    $true )).

fof(t7_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_series_1(A)
       => ( v4_seq_2(A)
          & k2_seq_2(A) = np__0 ) ) ) )).

fof(t8_series_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) )
         => k9_seq_1(k1_series_1(A),k1_series_1(B)) = k1_series_1(k9_seq_1(A,B)) ) ) )).

fof(t9_series_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(k1_series_1(A),k1_series_1(B)) = k1_series_1(k10_seq_1(A,B)) ) ) )).

fof(t10_series_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) )
         => ( ( v1_series_1(A)
              & v1_series_1(B) )
           => ( v1_series_1(k9_seq_1(A,B))
              & k2_series_1(k9_seq_1(A,B)) = k3_real_1(k2_series_1(A),k2_series_1(B)) ) ) ) ) )).

fof(t11_series_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) )
         => ( ( v1_series_1(A)
              & v1_series_1(B) )
           => ( v1_series_1(k10_seq_1(A,B))
              & k2_series_1(k10_seq_1(A,B)) = k5_real_1(k2_series_1(A),k2_series_1(B)) ) ) ) ) )).

fof(t12_series_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) )
         => k1_series_1(k14_seq_1(B,A)) = k14_seq_1(k1_series_1(B),A) ) ) )).

fof(t13_series_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) )
         => ( v1_series_1(B)
           => ( v1_series_1(k14_seq_1(B,A))
              & k2_series_1(k14_seq_1(B,A)) = k3_xcmplx_0(A,k2_series_1(B)) ) ) ) ) )).

fof(t14_series_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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,C) = k8_funct_2(k5_numbers,k1_numbers,A,np__0) )
           => k1_series_1(k1_seqm_3(A,np__1)) = k10_seq_1(k1_seqm_3(k1_series_1(A),np__1),B) ) ) ) )).

fof(t15_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_series_1(A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => v1_series_1(k1_seqm_3(A,B)) ) ) ) )).

fof(t16_series_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)
            & v1_series_1(k1_seqm_3(A,B)) )
       => v1_series_1(A) ) ) )).

fof(t17_series_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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,C),k8_funct_2(k5_numbers,k1_numbers,B,C)) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),C),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(B),C)) ) ) ) ) )).

fof(t18_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_series_1(A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_series_1(A) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),B),k2_series_1(k1_seqm_3(A,k1_nat_1(B,np__1)))) ) ) ) )).

fof(t19_series_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)
           => r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,B)) )
       => v3_seqm_3(k1_series_1(A)) ) ) )).

fof(t20_series_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)
           => r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,B)) )
       => ( v1_seq_2(k1_series_1(A))
        <=> v1_series_1(A) ) ) ) )).

fof(t21_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v1_series_1(A)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,B)) ) )
       => r1_xreal_0(np__0,k2_series_1(A)) ) ) )).

fof(t22_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C)) )
              & v1_series_1(B) )
           => ( ! [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(k8_funct_2(k5_numbers,k1_numbers,A,D),k8_funct_2(k5_numbers,k1_numbers,B,D)) ) )
              | v1_series_1(A) ) ) ) ) )).

fof(t23_series_1,axiom,(
    $true )).

fof(t24_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
                    & r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,C),k8_funct_2(k5_numbers,k1_numbers,B,C)) ) )
              & v1_series_1(B) )
           => ( v1_series_1(A)
              & r1_xreal_0(k2_series_1(A),k2_series_1(B)) ) ) ) ) )).

fof(t25_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_series_1(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,k1_series_1(A),D),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),C)))) ) ) ) ) ) ) )).

fof(t26_series_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( B != np__1
           => k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k2_prepower(B)),A) = k7_xcmplx_0(k6_xcmplx_0(np__1,k3_power(B,k1_nat_1(A,np__1))),k6_xcmplx_0(np__1,B)) ) ) ) )).

fof(t27_series_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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) = k3_xcmplx_0(A,k8_funct_2(k5_numbers,k1_numbers,B,C)) )
           => ( A = np__1
              | ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k8_funct_2(k5_numbers,k1_numbers,k1_series_1(B),C) = k7_xcmplx_0(k3_xcmplx_0(k8_funct_2(k5_numbers,k1_numbers,B,np__0),k6_xcmplx_0(np__1,k3_power(A,k1_nat_1(C,np__1)))),k6_xcmplx_0(np__1,A)) ) ) ) ) ) )).

fof(t28_series_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(np__1,k18_complex1(A))
       => ( v1_series_1(k2_prepower(A))
          & k2_series_1(k2_prepower(A)) = k7_xcmplx_0(np__1,k6_xcmplx_0(np__1,A)) ) ) ) )).

fof(t29_series_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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) = k3_xcmplx_0(A,k8_funct_2(k5_numbers,k1_numbers,B,C)) )
           => ( r1_xreal_0(np__1,k18_complex1(A))
              | ( v1_series_1(B)
                & k2_series_1(B) = k7_xcmplx_0(k8_funct_2(k5_numbers,k1_numbers,B,np__0),k6_xcmplx_0(np__1,A)) ) ) ) ) ) )).

fof(t30_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ~ r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,C),np__0)
                    & k8_funct_2(k5_numbers,k1_numbers,B,C) = k6_real_1(k8_funct_2(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,k1_numbers,A,C)) ) )
              & v4_seq_2(B) )
           => ( r1_xreal_0(np__1,k2_seq_2(B))
              | v1_series_1(A) ) ) ) ) )).

fof(t31_series_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)
             => ~ r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,B),np__0) )
          & ? [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(B,C)
                   => r1_xreal_0(np__1,k6_real_1(k8_funct_2(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,k1_numbers,A,C))) ) ) )
          & v1_series_1(A) ) ) )).

fof(t32_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
                    & k8_funct_2(k5_numbers,k1_numbers,B,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,A,C)) ) )
              & v4_seq_2(B) )
           => ( r1_xreal_0(np__1,k2_seq_2(B))
              | v1_series_1(A) ) ) ) ) )).

fof(t33_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
                    & k8_funct_2(k5_numbers,k1_numbers,B,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,A,C)) ) )
              & ? [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(np__1,k8_funct_2(k5_numbers,k1_numbers,B,D)) ) ) )
              & v1_series_1(A) ) ) ) )).

fof(t34_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
                    & k8_funct_2(k5_numbers,k1_numbers,B,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,A,C)) ) )
              & v4_seq_2(B)
              & ~ r1_xreal_0(k2_seq_2(B),np__1)
              & v1_series_1(A) ) ) ) )).

fof(t35_series_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) )
         => ( ( v4_seqm_3(A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
                    & k8_funct_2(k5_numbers,k1_numbers,B,C) = k4_real_1(k3_series_1(np__2,C),k8_funct_2(k5_numbers,k1_numbers,A,k3_series_1(np__2,C))) ) ) )
           => ( v1_series_1(A)
            <=> v1_series_1(B) ) ) ) ) )).

fof(t36_series_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] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(np__1,C)
                 => k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k3_power(C,A)) ) )
           => ( r1_xreal_0(A,np__1)
              | v1_series_1(B) ) ) ) ) )).

fof(t37_series_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) )
         => ~ ( r1_xreal_0(A,np__1)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(np__1,C)
                   => k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k3_power(C,A)) ) )
              & v1_series_1(B) ) ) ) )).

fof(d4_series_1,axiom,(
    $true )).

fof(d5_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_series_1(A)
      <=> v1_series_1(k22_seq_1(A)) ) ) )).

fof(t38_series_1,axiom,(
    $true )).

fof(t39_series_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)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(B,C)
               => r1_xreal_0(k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),C),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),B))),k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k22_seq_1(A)),C),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k22_seq_1(A)),B)))) ) ) ) ) )).

fof(t40_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_series_1(A)
       => v1_series_1(A) ) ) )).

fof(t41_series_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)
             => r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,B)) )
          & v1_series_1(A) )
       => v2_series_1(A) ) ) )).

fof(t42_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( k8_funct_2(k5_numbers,k1_numbers,A,C) != np__0
                    & k8_funct_2(k5_numbers,k1_numbers,B,C) = k6_real_1(k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(A),k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(A),C)) ) )
              & v4_seq_2(B) )
           => ( r1_xreal_0(np__1,k2_seq_2(B))
              | v2_series_1(A) ) ) ) ) )).

fof(t43_series_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) )
         => ~ ( ~ r1_xreal_0(A,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(A,k18_complex1(k8_funct_2(k5_numbers,k1_numbers,B,D))) ) ) )
              & v4_seq_2(B)
              & k2_seq_2(B) = np__0 ) ) ) )).

fof(t44_series_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)
             => k8_funct_2(k5_numbers,k1_numbers,A,B) != np__0 )
          & ? [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(B,C)
                   => r1_xreal_0(np__1,k6_real_1(k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(A),k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(A),C))) ) ) )
          & v1_series_1(A) ) ) )).

fof(t45_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k8_funct_2(k5_numbers,k1_numbers,A,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(B),C)) )
              & v4_seq_2(A) )
           => ( r1_xreal_0(np__1,k2_seq_2(A))
              | v2_series_1(B) ) ) ) ) )).

fof(t46_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k8_funct_2(k5_numbers,k1_numbers,A,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(B),C)) )
              & ? [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(np__1,k8_funct_2(k5_numbers,k1_numbers,A,D)) ) ) )
              & v1_series_1(B) ) ) ) )).

fof(t47_series_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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k8_funct_2(k5_numbers,k1_numbers,A,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(B),C)) )
              & v4_seq_2(A)
              & ~ r1_xreal_0(k2_seq_2(A),np__1)
              & v1_series_1(B) ) ) ) )).

fof(dt_k1_series_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(k1_series_1(A))
        & v1_funct_2(k1_series_1(A),k5_numbers,k1_numbers)
        & m2_relset_1(k1_series_1(A),k5_numbers,k1_numbers) ) ) )).

fof(dt_k2_series_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => m1_subset_1(k2_series_1(A),k1_numbers) ) )).

fof(dt_k3_series_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k3_series_1(A,B),k1_numbers,k5_numbers) ) )).

fof(redefinition_k3_series_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k3_series_1(A,B) = k3_power(A,B) ) )).
%------------------------------------------------------------------------------