SET007 Axioms: SET007+908.ax


%------------------------------------------------------------------------------
% File     : SET007+908 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Partial Product of Series and Related Basic Inequalities
% 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   55 (   0 unit)
%            Number of atoms       :  391 (  24 equality)
%            Maximal formula depth :   18 (   8 average)
%            Number of connectives :  357 (  21 ~  ;   4  |; 147  &)
%                                         (   1 <=>; 184 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   0 propositional; 1-3 arity)
%            Number of functors    :   36 (  10 constant; 0-4 arity)
%            Number of variables   :  156 (   0 singleton; 156 !;   0 ?)
%            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(fc1_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k16_complex1(A))
        & v1_xreal_0(k16_complex1(A))
        & ~ v3_xreal_0(k16_complex1(A)) ) ) )).

fof(t1_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(np__0,B)
                  & r1_xreal_0(np__0,C) )
               => ( r1_xreal_0(A,B)
                  | r1_xreal_0(k7_xcmplx_0(B,A),k7_xcmplx_0(k2_xcmplx_0(B,C),k2_xcmplx_0(A,C))) ) ) ) ) ) )).

fof(t2_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(k8_square_1(k3_xcmplx_0(A,B)),k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)) ) ) )).

fof(t3_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(np__2,k2_xcmplx_0(k7_xcmplx_0(A,B),k7_xcmplx_0(B,A))) ) ) )).

fof(t4_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k3_xcmplx_0(A,B),k5_square_1(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2))) ) ) )).

fof(t5_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k5_square_1(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)),k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)) ) ) )).

fof(t6_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(np__2,A),B),k2_xcmplx_0(k5_square_1(A),k5_square_1(B))) ) ) )).

fof(t7_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k3_xcmplx_0(A,B),k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)) ) ) )).

fof(t8_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k11_binop_2(k11_binop_2(np__2,k18_complex1(A)),k18_complex1(B)),k2_xcmplx_0(k5_square_1(A),k5_square_1(B))) ) ) )).

fof(t9_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(np__4,A),B),k5_square_1(k2_xcmplx_0(A,B))) ) ) )).

fof(t10_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(B,C)),k3_xcmplx_0(A,C)),k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C))) ) ) ) )).

fof(t11_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => r1_xreal_0(k3_xcmplx_0(np__3,k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(B,C)),k3_xcmplx_0(A,C))),k5_square_1(k2_xcmplx_0(k2_xcmplx_0(A,B),C))) ) ) ) )).

fof(t12_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & v2_xreal_0(C) )
             => r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__3,A),B),C),k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k2_newton(B,np__3)),k2_newton(C,np__3))) ) ) ) )).

fof(t13_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & v2_xreal_0(C) )
             => r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(A,B),C),k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k2_newton(B,np__3)),k2_newton(C,np__3)),np__3)) ) ) ) )).

fof(t14_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & v2_xreal_0(C) )
             => r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(B,A),k7_xcmplx_0(C,B)),k7_xcmplx_0(A,C)),k2_xcmplx_0(k2_xcmplx_0(k2_newton(k7_xcmplx_0(A,B),np__3),k2_newton(k7_xcmplx_0(B,C),np__3)),k2_newton(k7_xcmplx_0(C,A),np__3))) ) ) ) )).

fof(t15_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & v2_xreal_0(C) )
             => r1_xreal_0(k3_xcmplx_0(np__3,k1_power(np__3,k3_xcmplx_0(k3_xcmplx_0(A,B),C))),k2_xcmplx_0(k2_xcmplx_0(A,B),C)) ) ) ) )).

fof(t16_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & v2_xreal_0(C) )
             => r1_xreal_0(k1_power(np__3,k3_xcmplx_0(k3_xcmplx_0(A,B),C)),k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(A,B),C),np__3)) ) ) ) )).

fof(t17_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
               => r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(B,C)),k3_xcmplx_0(A,C)),k12_binop_2(np__1,np__3)) ) ) ) ) )).

fof(t18_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( k2_xcmplx_0(A,B) = np__1
           => r1_xreal_0(k3_xcmplx_0(A,B),k12_binop_2(np__1,np__4)) ) ) ) )).

fof(t19_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( k2_xcmplx_0(A,B) = np__1
           => r1_xreal_0(k12_binop_2(np__1,np__2),k2_xcmplx_0(k5_square_1(A),k5_square_1(B))) ) ) ) )).

fof(t20_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ( k2_xcmplx_0(A,B) = np__1
           => r1_xreal_0(np__9,k3_xcmplx_0(k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,A)),k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,B)))) ) ) ) )).

fof(t21_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( k2_xcmplx_0(A,B) = np__1
           => r1_xreal_0(k12_binop_2(np__1,np__4),k2_xcmplx_0(k2_newton(A,np__3),k2_newton(B,np__3))) ) ) ) )).

fof(t22_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ~ ( k2_xcmplx_0(A,B) = np__1
              & r1_xreal_0(np__1,k2_xcmplx_0(k2_newton(A,np__3),k2_newton(B,np__3))) ) ) ) )).

fof(t23_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ( k2_xcmplx_0(A,B) = np__1
           => r1_xreal_0(k12_binop_2(np__25,np__4),k3_xcmplx_0(k2_xcmplx_0(A,k7_xcmplx_0(np__1,A)),k2_xcmplx_0(B,k7_xcmplx_0(np__1,B)))) ) ) ) )).

fof(t24_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(k18_complex1(B),A)
           => r1_xreal_0(k5_square_1(B),k5_square_1(A)) ) ) ) )).

fof(t25_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,k18_complex1(B))
           => r1_xreal_0(k5_square_1(A),k5_square_1(B)) ) ) ) )).

fof(t26_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k18_complex1(k10_binop_2(k18_complex1(A),k18_complex1(B))),k9_binop_2(k18_complex1(A),k18_complex1(B))) ) ) )).

fof(t27_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & v2_xreal_0(C) )
             => ( k3_xcmplx_0(k3_xcmplx_0(A,B),C) = np__1
               => r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k8_square_1(A),k8_square_1(B)),k8_square_1(C)),k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B)),k7_xcmplx_0(np__1,C))) ) ) ) ) )).

fof(t28_series_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__0
               => ( r1_xreal_0(A,np__0)
                  | r1_xreal_0(B,np__0)
                  | r1_xreal_0(np__0,C)
                  | r1_xreal_0(k3_xcmplx_0(np__6,k5_square_1(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k2_newton(B,np__3)),k2_newton(C,np__3)))),k2_newton(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__2),k2_newton(B,np__2)),k2_newton(C,np__2)),np__3)) ) ) ) ) ) )).

fof(t29_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & v2_xreal_0(C) )
             => ( r1_xreal_0(np__1,A)
               => r1_xreal_0(k3_xcmplx_0(np__2,k3_power(A,k8_square_1(k3_xcmplx_0(B,C)))),k2_xcmplx_0(k3_power(A,B),k3_power(A,C))) ) ) ) ) )).

fof(t30_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & v2_xreal_0(C) )
             => ( ( r1_xreal_0(B,A)
                  & r1_xreal_0(C,B) )
               => r1_xreal_0(k3_power(k3_xcmplx_0(k3_xcmplx_0(A,B),C),k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(A,B),C),np__3)),k3_xcmplx_0(k3_xcmplx_0(k3_power(A,A),k3_power(B,B)),k3_power(C,C))) ) ) ) ) )).

fof(t31_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => r1_xreal_0(k2_xcmplx_0(k2_newton(A,k23_binop_2(C,np__2)),k3_xcmplx_0(k3_xcmplx_0(k23_binop_2(C,np__2),k2_newton(A,k23_binop_2(C,np__1))),B)),k2_newton(k2_xcmplx_0(A,B),k23_binop_2(C,np__2))) ) ) ) )).

fof(t32_series_3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => r1_xreal_0(k2_newton(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2),C),k7_xcmplx_0(k2_xcmplx_0(k2_newton(A,C),k2_newton(B,C)),np__2)) ) ) ) )).

fof(t33_series_3,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(k2_seq_1(k5_numbers,k1_numbers,A,B),np__0) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),np__0) ) ) ) )).

fof(t34_series_3,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,k2_seq_1(k5_numbers,k1_numbers,A,B)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) ) ) )).

fof(t35_series_3,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) )
         => ~ ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,C)) )
              & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A)) ) ) ) )).

fof(t36_series_3,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 = k11_seq_1(B,B)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),C)) ) ) ) ) )).

fof(t37_series_3,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) )
         => ~ ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),np__0)
                    & ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),k2_seq_1(k5_numbers,k1_numbers,B,k10_binop_2(C,np__1))) ) )
              & r1_xreal_0(k11_binop_2(k23_binop_2(A,np__1),k2_seq_1(k5_numbers,k1_numbers,B,k23_binop_2(A,np__1))),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A)) ) ) ) )).

fof(t38_series_3,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 = k11_seq_1(B,C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,D))
                        & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) )
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),D),k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),D),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(C),D))) ) ) ) ) ) )).

fof(t39_series_3,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) )
         => ! [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) )
                 => ( ( B = k11_seq_1(C,D)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,E))
                            & ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,E)) ) ) )
                   => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A),k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(C),A),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(D),A))) ) ) ) ) ) )).

fof(t40_series_3,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(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k22_seq_1(A)),B)) ) ) )).

fof(t41_series_3,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) )
         => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k22_seq_1(B)),A)) ) ) )).

fof(d1_series_3,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_3(A)
          <=> ( k2_seq_1(k5_numbers,k1_numbers,B,np__0) = k2_seq_1(k5_numbers,k1_numbers,A,np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k2_seq_1(k5_numbers,k1_numbers,B,k23_binop_2(C,np__1)) = k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,B,C),k2_seq_1(k5_numbers,k1_numbers,A,k23_binop_2(C,np__1))) ) ) ) ) ) )).

fof(t42_series_3,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) )
         => ~ ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),np__0) )
              & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),A),np__0) ) ) ) )).

fof(t43_series_3,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) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,C)) )
           => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),A)) ) ) ) )).

fof(t44_series_3,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(k2_seq_1(k5_numbers,k1_numbers,A,B),np__0)
              & ~ r1_xreal_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B),np__0)
              & ~ r1_xreal_0(np__1,k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B)) ) ) ) ) )).

fof(t45_series_3,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__1,k2_seq_1(k5_numbers,k1_numbers,A,B)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r1_xreal_0(np__1,k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B)) ) ) ) )).

fof(t46_series_3,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,k2_seq_1(k5_numbers,k1_numbers,A,C))
                  & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,C)) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_xreal_0(k9_binop_2(k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),C),k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),C)),k2_seq_1(k5_numbers,k1_numbers,k1_series_3(k9_seq_1(A,B)),C)) ) ) ) ) )).

fof(t47_series_3,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) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_seq_1(k5_numbers,k1_numbers,B,C) = k12_binop_2(k23_binop_2(k24_binop_2(np__2,C),np__1),k23_binop_2(k24_binop_2(np__2,C),np__2)) )
           => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),A),k12_binop_2(np__1,k9_square_1(k23_binop_2(k24_binop_2(np__3,A),np__4)))) ) ) ) )).

fof(t48_series_3,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)
               => ( k2_seq_1(k5_numbers,k1_numbers,A,C) = k9_binop_2(np__1,k2_seq_1(k5_numbers,k1_numbers,B,C))
                  & ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),k7_binop_2(np__1))
                  & ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,C)) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_xreal_0(k9_binop_2(np__1,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C)),k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),C)) ) ) ) ) )).

fof(t49_series_3,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)
               => ( k2_seq_1(k5_numbers,k1_numbers,A,C) = k9_binop_2(np__1,k2_seq_1(k5_numbers,k1_numbers,B,C))
                  & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,C)) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_xreal_0(k9_binop_2(np__1,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C)),k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),C)) ) ) ) ) )).

fof(t50_series_3,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) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k5_numbers,k1_numbers)
                        & m2_relset_1(E,k5_numbers,k1_numbers) )
                     => ( ( A = k11_seq_1(B,C)
                          & D = k11_seq_1(B,B)
                          & E = k11_seq_1(C,C) )
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => r1_xreal_0(k7_square_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),F)),k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(D),F),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(E),F))) ) ) ) ) ) ) ) )).

fof(t51_series_3,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) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k5_numbers,k1_numbers)
                        & m2_relset_1(E,k5_numbers,k1_numbers) )
                     => ( ( A = k11_seq_1(B,B)
                          & C = k11_seq_1(D,D)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,F))
                                & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F))
                                & k2_seq_1(k5_numbers,k1_numbers,E,F) = k7_square_1(k9_binop_2(k2_seq_1(k5_numbers,k1_numbers,B,F),k2_seq_1(k5_numbers,k1_numbers,D,F))) ) ) )
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => r1_xreal_0(k9_square_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(E),F)),k9_binop_2(k9_square_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),F)),k9_square_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(C),F)))) ) ) ) ) ) ) ) )).

fof(t52_series_3,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) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),np__0)
                  & ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),k2_seq_1(k5_numbers,k1_numbers,B,k10_binop_2(C,np__1))) ) )
           => r1_xreal_0(k11_binop_2(k23_binop_2(A,np__1),k2_power(k23_binop_2(A,np__1),k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),A))),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A)) ) ) ) )).

fof(dt_k1_series_3,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_3(A))
        & v1_funct_2(k1_series_3(A),k5_numbers,k1_numbers)
        & m2_relset_1(k1_series_3(A),k5_numbers,k1_numbers) ) ) )).
%------------------------------------------------------------------------------