SET007 Axioms: SET007+926.ax


%------------------------------------------------------------------------------
% File     : SET007+926 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Partial Product and Sum 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_5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   58 (   0 unit)
%            Number of atoms       :  415 (  23 equality)
%            Maximal formula depth :   13 (   8 average)
%            Number of connectives :  403 (  46 ~  ;   0  |; 163  &)
%                                         (   0 <=>; 194 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   0 propositional; 1-3 arity)
%            Number of functors    :   28 (  12 constant; 0-4 arity)
%            Number of variables   :  169 (   0 singleton; 169 !;   0 ?)
%            Maximal term depth    :    8 (   2 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_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(np__4,k3_xcmplx_0(k2_xcmplx_0(A,B),k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B)))) ) ) )).

fof(t2_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(k2_xcmplx_0(k3_xcmplx_0(k2_newton(A,np__3),B),k3_xcmplx_0(A,k2_newton(B,np__3))),k2_xcmplx_0(k2_newton(A,np__4),k2_newton(B,np__4))) ) ) )).

fof(t3_series_5,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(k7_xcmplx_0(k2_xcmplx_0(B,C),k2_xcmplx_0(A,C)),np__1) ) ) ) ) )).

fof(t4_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ~ ( ~ r1_xreal_0(B,A)
              & r1_xreal_0(k8_square_1(k7_xcmplx_0(A,B)),k7_xcmplx_0(A,B)) ) ) ) )).

fof(t5_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ~ ( ~ r1_xreal_0(B,A)
              & r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(B,k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))),k2_xcmplx_0(A,k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)))),k8_square_1(k7_xcmplx_0(A,B))) ) ) ) )).

fof(t6_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ~ ( ~ r1_xreal_0(B,A)
              & r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(B,k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))),k2_xcmplx_0(A,k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)))),k7_xcmplx_0(A,B)) ) ) ) )).

fof(t7_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(k7_xcmplx_0(np__2,k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B))),k8_square_1(k3_xcmplx_0(A,B))) ) ) )).

fof(t8_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))) ) ) )).

fof(t9_series_5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k2_xcmplx_0(A,B),k8_square_1(k3_xcmplx_0(np__2,k2_xcmplx_0(k5_square_1(A),k5_square_1(B))))) ) ) )).

fof(t10_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(k7_xcmplx_0(np__2,k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B))),k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)) ) ) )).

fof(t11_series_5,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)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))) ) ) )).

fof(t12_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(k7_xcmplx_0(np__2,k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))) ) ) )).

fof(t13_series_5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(np__1,k18_complex1(A))
              & ~ r1_xreal_0(np__1,k18_complex1(B))
              & ~ r1_xreal_0(k18_complex1(k7_xcmplx_0(k2_xcmplx_0(A,B),k2_xcmplx_0(np__1,k3_xcmplx_0(A,B)))),np__1) ) ) ) )).

fof(t14_series_5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k7_xcmplx_0(k18_complex1(k2_xcmplx_0(A,B)),k2_xcmplx_0(np__1,k18_complex1(k2_xcmplx_0(A,B)))),k2_xcmplx_0(k7_xcmplx_0(k18_complex1(A),k2_xcmplx_0(np__1,k18_complex1(A))),k7_xcmplx_0(k18_complex1(B),k2_xcmplx_0(np__1,k18_complex1(B))))) ) ) )).

fof(t15_series_5,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) )
             => ! [D] :
                  ( ( v1_xreal_0(D)
                    & v2_xreal_0(D) )
                 => ~ r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(A,k2_xcmplx_0(k2_xcmplx_0(A,B),C)),k7_xcmplx_0(B,k2_xcmplx_0(k2_xcmplx_0(A,B),D))),k7_xcmplx_0(D,k2_xcmplx_0(k2_xcmplx_0(B,D),C))),k7_xcmplx_0(C,k2_xcmplx_0(k2_xcmplx_0(A,D),C))),np__1) ) ) ) ) )).

fof(t16_series_5,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) )
             => ! [D] :
                  ( ( v1_xreal_0(D)
                    & v2_xreal_0(D) )
                 => ~ r1_xreal_0(np__2,k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(A,k2_xcmplx_0(k2_xcmplx_0(A,B),C)),k7_xcmplx_0(B,k2_xcmplx_0(k2_xcmplx_0(A,B),D))),k7_xcmplx_0(D,k2_xcmplx_0(k2_xcmplx_0(B,D),C))),k7_xcmplx_0(C,k2_xcmplx_0(k2_xcmplx_0(A,D),C)))) ) ) ) ) )).

fof(t17_series_5,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(A,B),C)
                  & ~ r1_xreal_0(k2_xcmplx_0(B,C),A)
                  & ~ r1_xreal_0(k2_xcmplx_0(A,C),B)
                  & ~ r1_xreal_0(k7_xcmplx_0(np__9,k2_xcmplx_0(k2_xcmplx_0(A,B),C)),k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(np__1,k6_xcmplx_0(k2_xcmplx_0(A,B),C)),k7_xcmplx_0(np__1,k6_xcmplx_0(k2_xcmplx_0(B,C),A))),k7_xcmplx_0(np__1,k6_xcmplx_0(k2_xcmplx_0(C,A),B)))) ) ) ) ) )).

fof(t18_series_5,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) )
             => ! [D] :
                  ( ( v1_xreal_0(D)
                    & v2_xreal_0(D) )
                 => r1_xreal_0(k2_xcmplx_0(k8_square_1(k3_xcmplx_0(A,C)),k8_square_1(k3_xcmplx_0(B,D))),k8_square_1(k3_xcmplx_0(k2_xcmplx_0(A,B),k2_xcmplx_0(C,D)))) ) ) ) ) )).

fof(t19_series_5,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) )
             => ! [D] :
                  ( ( v1_xreal_0(D)
                    & v2_xreal_0(D) )
                 => r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__4,A),B),C),D),k3_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(C,D)),k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)))) ) ) ) ) )).

fof(t20_series_5,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__3,k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(A,B),k7_xcmplx_0(B,C)),k7_xcmplx_0(C,A))) ) ) ) )).

fof(t21_series_5,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) )
             => ( k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(B,C)),k3_xcmplx_0(C,A)) = np__1
               => r1_xreal_0(k9_square_1(np__3),k2_xcmplx_0(k2_xcmplx_0(A,B),C)) ) ) ) ) )).

fof(t22_series_5,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__3,k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(A,B),C),C),k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(B,C),A),A)),k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(C,A),B),B))) ) ) ) )).

fof(t23_series_5,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => r1_xreal_0(k5_square_1(k2_xcmplx_0(k8_square_1(k3_xcmplx_0(A,B)),k7_xcmplx_0(np__1,k8_square_1(k3_xcmplx_0(A,B))))),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_5,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(C,A),B),k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(k3_xcmplx_0(A,B),C),k7_xcmplx_0(k3_xcmplx_0(C,B),A)),k7_xcmplx_0(k3_xcmplx_0(C,A),B))) ) ) ) )).

fof(t25_series_5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,B)
                  & ~ r1_xreal_0(B,C)
                  & r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k5_square_1(A),B),k3_xcmplx_0(k5_square_1(B),C)),k3_xcmplx_0(k5_square_1(C),A)),k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,k5_square_1(B)),k3_xcmplx_0(B,k5_square_1(C))),k3_xcmplx_0(C,k5_square_1(A)))) ) ) ) ) )).

fof(t26_series_5,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(A,B)
                  & ~ r1_xreal_0(B,C)
                  & r1_xreal_0(k7_xcmplx_0(B,k6_xcmplx_0(A,B)),k7_xcmplx_0(C,k6_xcmplx_0(A,C))) ) ) ) ) )).

fof(t27_series_5,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) )
             => ! [D] :
                  ( ( v1_xreal_0(D)
                    & v2_xreal_0(D) )
                 => ~ ( ~ r1_xreal_0(A,B)
                      & ~ r1_xreal_0(C,D)
                      & r1_xreal_0(k7_xcmplx_0(C,k2_xcmplx_0(C,B)),k7_xcmplx_0(D,k2_xcmplx_0(D,A))) ) ) ) ) ) )).

fof(t28_series_5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => r1_xreal_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(C,D)),k3_xcmplx_0(k8_square_1(k2_xcmplx_0(k5_square_1(A),k5_square_1(C))),k8_square_1(k2_xcmplx_0(k5_square_1(B),k5_square_1(D))))) ) ) ) ) )).

fof(t29_series_5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ! [E] :
                      ( v1_xreal_0(E)
                     => ! [F] :
                          ( v1_xreal_0(F)
                         => r1_xreal_0(k5_square_1(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(C,D)),k3_xcmplx_0(E,F))),k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(C)),k5_square_1(E)),k2_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k5_square_1(D)),k5_square_1(F)))) ) ) ) ) ) ) )).

fof(t30_series_5,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(k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__9,A),B),C),k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C))),k2_xcmplx_0(k2_xcmplx_0(A,B),C)) ) ) ) )).

fof(t31_series_5,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(A,B),C),k2_xcmplx_0(k2_xcmplx_0(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k3_xcmplx_0(A,B)),k5_square_1(B)),np__3)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k3_xcmplx_0(B,C)),k5_square_1(C)),np__3))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k3_xcmplx_0(C,A)),k5_square_1(A)),np__3)))) ) ) ) )).

fof(t32_series_5,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(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k3_xcmplx_0(A,B)),k5_square_1(B)),np__3)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k3_xcmplx_0(B,C)),k5_square_1(C)),np__3))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k3_xcmplx_0(C,A)),k5_square_1(A)),np__3))),k2_xcmplx_0(k2_xcmplx_0(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k5_square_1(C)),np__2))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k5_square_1(A)),np__2)))) ) ) ) )).

fof(t33_series_5,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(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k5_square_1(C)),np__2))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k5_square_1(A)),np__2))),k8_square_1(k3_xcmplx_0(np__3,k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C))))) ) ) ) )).

fof(t34_series_5,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(k8_square_1(k3_xcmplx_0(np__3,k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C)))),k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(k3_xcmplx_0(B,C),A),k7_xcmplx_0(k3_xcmplx_0(C,A),B)),k7_xcmplx_0(k3_xcmplx_0(A,B),C))) ) ) ) )).

fof(t35_series_5,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(k6_xcmplx_0(k7_xcmplx_0(np__1,k5_square_1(A)),np__1),k6_xcmplx_0(k7_xcmplx_0(np__1,k5_square_1(B)),np__1))) ) ) ) )).

fof(t36_series_5,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(k7_xcmplx_0(np__17,np__4),k2_xcmplx_0(k3_xcmplx_0(A,B),k7_xcmplx_0(np__1,k3_xcmplx_0(A,B)))) ) ) ) )).

fof(t37_series_5,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) )
             => ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
               => r1_xreal_0(np__9,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(t38_series_5,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) )
             => ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
               => r1_xreal_0(np__8,k3_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__1,A),np__1),k6_xcmplx_0(k7_xcmplx_0(np__1,B),np__1)),k6_xcmplx_0(k7_xcmplx_0(np__1,C),np__1))) ) ) ) ) )).

fof(t39_series_5,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) )
             => ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
               => r1_xreal_0(np__64,k3_xcmplx_0(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))),k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,C)))) ) ) ) ) )).

fof(t40_series_5,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(k7_xcmplx_0(np__1,np__3),k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C))) ) ) ) ) )).

fof(t41_series_5,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(C,A)),k7_xcmplx_0(np__1,np__3)) ) ) ) ) )).

fof(t42_series_5,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(t43_series_5,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(A,B)
                  & ~ r1_xreal_0(B,C)
                  & r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(k3_power(A,k3_xcmplx_0(np__2,A)),k3_power(B,k3_xcmplx_0(np__2,B))),k3_power(C,k3_xcmplx_0(np__2,C))),k3_xcmplx_0(k3_xcmplx_0(k3_power(A,k2_xcmplx_0(B,C)),k3_power(B,k2_xcmplx_0(A,C))),k3_power(C,k2_xcmplx_0(A,B)))) ) ) ) ) )).

fof(t44_series_5,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(np__1,C)
               => r1_xreal_0(k2_xcmplx_0(k3_xcmplx_0(k2_newton(A,C),B),k3_xcmplx_0(A,k2_newton(B,C))),k2_xcmplx_0(k2_newton(A,k1_nat_1(C,np__1)),k2_newton(B,k1_nat_1(C,np__1)))) ) ) ) ) )).

fof(t45_series_5,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) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( k2_xcmplx_0(k5_square_1(A),k5_square_1(B)) = k5_square_1(C)
                      & r1_xreal_0(np__3,D)
                      & r1_xreal_0(k2_newton(C,k1_nat_1(D,np__2)),k2_xcmplx_0(k2_newton(A,k1_nat_1(D,np__2)),k2_newton(B,k1_nat_1(D,np__2)))) ) ) ) ) ) )).

fof(t46_series_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( r1_xreal_0(np__1,A)
          & r1_xreal_0(k2_newton(k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,A)),k1_nat_1(A,np__1)),k2_newton(k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,k1_nat_1(A,np__1))),A)) ) ) )).

fof(t47_series_5,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)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(np__1,C)
                      & r1_xreal_0(np__1,D) )
                   => r1_xreal_0(k3_xcmplx_0(k2_xcmplx_0(k2_newton(A,D),k2_newton(B,D)),k2_xcmplx_0(k2_newton(A,C),k2_newton(B,C))),k3_xcmplx_0(np__2,k2_xcmplx_0(k2_newton(A,k1_nat_1(D,C)),k2_newton(B,k1_nat_1(D,C))))) ) ) ) ) ) )).

fof(t48_series_5,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) = k7_xcmplx_0(np__1,k9_square_1(k1_nat_1(B,np__1))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ r1_xreal_0(k3_xcmplx_0(np__2,k9_square_1(k1_nat_1(B,np__1))),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) ) ) )).

fof(t49_series_5,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) = k7_xcmplx_0(np__1,k7_square_1(k1_nat_1(B,np__1))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),k6_xcmplx_0(np__2,k7_xcmplx_0(np__1,k1_nat_1(B,np__1)))) ) ) ) )).

fof(t50_series_5,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) = k7_xcmplx_0(np__1,k7_square_1(k1_nat_1(C,np__1))) )
              & r1_xreal_0(np__2,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A)) ) ) ) )).

fof(t51_series_5,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(k1_nat_1(B,np__1),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) ) ) )).

fof(t52_series_5,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(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),B),k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B)) ) ) ) )).

fof(t53_series_5,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(k2_seq_1(k5_numbers,k1_numbers,A,C),np__0)
                  & k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C),np__0) ) ) ) ) )).

fof(t54_series_5,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(k2_seq_1(k5_numbers,k1_numbers,A,C),np__0)
                  & k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_xreal_0(k7_square_1(k1_nat_1(C,np__1)),k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),C),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C))) ) ) ) ) )).

fof(t55_series_5,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,B)
             => ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k9_square_1(B)
                & k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0 ) ) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( r1_xreal_0(np__1,B)
                & r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(np__1,np__6),k1_nat_1(k2_nat_1(np__4,B),np__3)),k9_square_1(B)),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) ) ) ) )).

fof(t56_series_5,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,B)
             => ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k9_square_1(B)
                & k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0 ) ) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( r1_xreal_0(np__1,B)
                & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),k3_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(np__2,np__3),B),k9_square_1(B))) ) ) ) ) )).

fof(t57_series_5,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,B)
             => ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,k1_nat_1(k2_nat_1(np__2,B),np__1)))
                & k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__1 ) ) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( r1_xreal_0(np__1,B)
                & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B),k3_xcmplx_0(k7_xcmplx_0(np__1,np__2),k9_square_1(k1_nat_1(k2_nat_1(np__2,B),np__3)))) ) ) ) ) )).

fof(t58_series_5,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,B)
             => ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k9_square_1(k2_nat_1(B,k1_nat_1(B,np__1)))
                & k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0 ) ) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( r1_xreal_0(np__1,B)
                & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),k7_xcmplx_0(k2_nat_1(B,k1_nat_1(B,np__1)),np__2)) ) ) ) ) )).
%------------------------------------------------------------------------------