SET007 Axioms: SET007+858.ax


%------------------------------------------------------------------------------
% File     : SET007+858 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Partial Sum of Some 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_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   42 (   0 unit)
%            Number of atoms       :  322 (  92 equality)
%            Maximal formula depth :   10 (   7 average)
%            Number of connectives :  280 (   0 ~  ;   0  |;  90  &)
%                                         (   0 <=>; 190 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    7 (   0 propositional; 1-3 arity)
%            Number of functors    :   31 (  18 constant; 0-4 arity)
%            Number of variables   :  126 (   0 singleton; 126 !;   0 ?)
%            Maximal term depth    :    9 (   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_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k18_complex1(k2_newton(k4_xcmplx_0(np__1),A)) = np__1 ) )).

fof(t2_series_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k2_newton(k2_xcmplx_0(A,np__1),np__3) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k3_xcmplx_0(np__3,k2_newton(A,np__2))),k3_xcmplx_0(np__3,A)),np__1)
        & k2_newton(k2_xcmplx_0(A,np__1),np__4) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__4),k3_xcmplx_0(np__4,k2_newton(A,np__3))),k3_xcmplx_0(np__6,k2_newton(A,np__2))),k3_xcmplx_0(np__4,A)),np__1)
        & k2_newton(k2_xcmplx_0(A,np__1),np__5) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__5),k3_xcmplx_0(np__5,k2_newton(A,np__4))),k3_xcmplx_0(np__10,k2_newton(A,np__3))),k3_xcmplx_0(np__10,k2_newton(A,np__2))),k3_xcmplx_0(np__5,A)),np__1) ) ) )).

fof(t3_series_2,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) = B )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => 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) ) ) ) )).

fof(t4_series_2,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) = k2_nat_1(np__2,B) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_nat_1(B,k1_nat_1(B,np__1)) ) ) ) )).

fof(t5_series_2,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) = k1_nat_1(k2_nat_1(np__2,B),np__1) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k3_newton(k1_nat_1(B,np__1),np__2) ) ) ) )).

fof(t6_series_2,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) = k2_nat_1(B,k1_nat_1(B,np__1)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),np__3) ) ) ) )).

fof(t7_series_2,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) = k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)),np__4) ) ) ) )).

fof(t8_series_2,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) = k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)),k1_nat_1(B,np__4)),np__5) ) ) ) )).

fof(t9_series_2,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,k2_nat_1(B,k1_nat_1(B,np__1))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k1_nat_1(B,np__1))) ) ) ) )).

fof(t10_series_2,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,k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__4),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(np__2,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)))) ) ) ) )).

fof(t11_series_2,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,k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__18),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(k2_nat_1(np__3,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)))) ) ) ) )).

fof(t12_series_2,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) = k3_newton(B,np__2) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(k2_nat_1(np__2,B),np__1)),np__6) ) ) ) )).

fof(t13_series_2,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) = k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1)),k3_newton(B,np__2)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1)),B),k1_nat_1(B,np__1)),np__2) ) ) ) )).

fof(t14_series_2,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_newton(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),np__2)
                & 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)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(B,k6_xcmplx_0(k3_xcmplx_0(np__4,k3_newton(B,np__2)),np__1)),np__3) ) ) ) ) )).

fof(t15_series_2,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) = k3_newton(B,np__3) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_newton(B,np__2),k3_newton(k1_nat_1(B,np__1),np__2)),np__4) ) ) ) )).

fof(t16_series_2,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_newton(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),np__3)
                & 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)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k3_xcmplx_0(k3_newton(B,np__2),k6_xcmplx_0(k3_xcmplx_0(np__2,k3_newton(B,np__2)),np__1)) ) ) ) ) )).

fof(t17_series_2,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) = k3_newton(B,np__4) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(k2_nat_1(np__2,B),np__1)),k6_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(np__3,k3_newton(B,np__2)),k2_nat_1(np__3,B)),np__1)),np__30) ) ) ) )).

fof(t18_series_2,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) = k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1)),k3_newton(B,np__4)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1)),B),k1_nat_1(B,np__1)),k6_xcmplx_0(k2_xcmplx_0(k3_newton(B,np__2),B),np__1)),np__2) ) ) ) )).

fof(t19_series_2,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) = k3_newton(B,np__5) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_newton(B,np__2),k3_newton(k1_nat_1(B,np__1),np__2)),k6_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(np__2,k3_newton(B,np__2)),k2_nat_1(np__2,B)),np__1)),np__12) ) ) ) )).

fof(t20_series_2,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) = k3_newton(B,np__6) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(k2_nat_1(np__2,B),np__1)),k2_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(np__3,k3_newton(B,np__4)),k3_xcmplx_0(np__6,k3_newton(B,np__3))),k2_nat_1(np__3,B)),np__1)),np__42) ) ) ) )).

fof(t21_series_2,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) = k3_newton(B,np__7) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_newton(B,np__2),k3_newton(k1_nat_1(B,np__1),np__2)),k2_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(np__3,k3_newton(B,np__4)),k3_xcmplx_0(np__6,k3_newton(B,np__3))),k3_newton(B,np__2)),k2_nat_1(np__4,B)),np__2)),np__24) ) ) ) )).

fof(t22_series_2,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) = k3_xcmplx_0(B,k3_newton(k1_nat_1(B,np__1),np__2)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(k2_nat_1(np__3,B),np__5)),np__12) ) ) ) )).

fof(t23_series_2,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) = k3_xcmplx_0(k3_xcmplx_0(B,k3_newton(k1_nat_1(B,np__1),np__2)),k1_nat_1(B,np__2)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)),k1_nat_1(k2_nat_1(np__2,B),np__3)),np__10) ) ) ) )).

fof(t24_series_2,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) = k3_xcmplx_0(k2_nat_1(B,k1_nat_1(B,np__1)),k3_newton(np__2,B)) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k3_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__1)),k2_xcmplx_0(k6_xcmplx_0(k3_newton(B,np__2),B),np__2)),np__4) ) ) ) )).

fof(t25_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k6_xcmplx_0(B,np__1),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__2,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__3,np__4),k7_xcmplx_0(np__1,k2_nat_1(np__2,B))),k7_xcmplx_0(np__1,k2_nat_1(np__2,k1_nat_1(B,np__1)))) ) ) ) ) )).

fof(t26_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),k1_nat_1(k2_nat_1(np__2,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)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(B,k1_nat_1(k2_nat_1(np__2,B),np__1)) ) ) ) ) )).

fof(t27_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__3,B),np__2),k1_nat_1(k2_nat_1(np__3,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)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(B,k1_nat_1(k2_nat_1(np__3,B),np__1)) ) ) ) ) )).

fof(t28_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),k1_nat_1(k2_nat_1(np__2,B),np__1)),k1_nat_1(k2_nat_1(np__2,B),np__3)))
                & 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)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__12),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(np__4,k1_nat_1(k2_nat_1(np__2,B),np__1)),k1_nat_1(k2_nat_1(np__2,B),np__3)))) ) ) ) ) )).

fof(t29_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__3,B),np__2),k1_nat_1(k2_nat_1(np__3,B),np__1)),k1_nat_1(k2_nat_1(np__3,B),np__4)))
                & 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)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__24),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(np__6,k1_nat_1(k2_nat_1(np__3,B),np__1)),k1_nat_1(k2_nat_1(np__3,B),np__4)))) ) ) ) ) )).

fof(t30_series_2,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(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__3,np__4),k7_xcmplx_0(np__2,k1_nat_1(B,np__2))),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(np__2,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)))) ) ) ) ) )).

fof(t31_series_2,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(k1_nat_1(B,np__2),k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__3))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__29,np__36),k7_xcmplx_0(np__1,k1_nat_1(B,np__3))),k7_xcmplx_0(np__3,k2_nat_1(k2_nat_1(np__2,k1_nat_1(B,np__2)),k1_nat_1(B,np__3)))),k7_xcmplx_0(np__4,k2_nat_1(k2_nat_1(k2_nat_1(np__3,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)))) ) ) ) ) )).

fof(t32_series_2,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(k3_xcmplx_0(k1_nat_1(B,np__1),k3_newton(np__2,B)),k2_nat_1(k1_nat_1(B,np__2),k1_nat_1(B,np__3))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__1)),k1_nat_1(B,np__3)),k7_xcmplx_0(np__1,np__2)) ) ) ) )).

fof(t33_series_2,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(k3_xcmplx_0(k3_newton(B,np__2),k3_newton(np__4,B)),k2_nat_1(k1_nat_1(B,np__1),k1_nat_1(B,np__2))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_xcmplx_0(k7_xcmplx_0(np__2,np__3),k7_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(B,np__1),k3_newton(np__4,k1_nat_1(B,np__1))),k2_nat_1(np__3,k1_nat_1(B,np__2)))) ) ) ) ) )).

fof(t34_series_2,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(k1_nat_1(B,np__2),k3_xcmplx_0(k2_nat_1(B,k1_nat_1(B,np__1)),k3_newton(np__2,B))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k3_xcmplx_0(k1_nat_1(B,np__1),k3_newton(np__2,B)))) ) ) ) ) )).

fof(t35_series_2,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(k1_nat_1(k2_nat_1(np__2,B),np__3),k3_xcmplx_0(k2_nat_1(B,k1_nat_1(B,np__1)),k3_newton(np__3,B))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k3_xcmplx_0(k1_nat_1(B,np__1),k3_newton(np__3,B)))) ) ) ) ) )).

fof(t36_series_2,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(k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),B),k3_newton(np__2,k1_nat_1(B,np__1))),k3_xcmplx_0(k2_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__1)),k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1))),k2_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__2)),k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__2))))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_xcmplx_0(k7_xcmplx_0(np__1,np__3),k7_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__2)),k3_xcmplx_0(np__3,k2_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__2)),k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__2)))))) ) ) ) )).

fof(t37_series_2,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) = k2_nat_1(k11_newton(B),B) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k11_newton(k1_nat_1(B,np__1)),np__1) ) ) ) ) )).

fof(t38_series_2,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(B,k11_newton(k1_nat_1(B,np__1))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k11_newton(k1_nat_1(B,np__1)))) ) ) ) ) )).

fof(t39_series_2,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) = k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k3_newton(B,np__2),B),np__1),k11_newton(k1_nat_1(B,np__2)))
                & 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)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__2),k7_xcmplx_0(k1_nat_1(B,np__1),k11_newton(k1_nat_1(B,np__2)))) ) ) ) ) )).

fof(t40_series_2,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(k3_xcmplx_0(B,k3_newton(np__2,B)),k11_newton(k1_nat_1(B,np__2))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__1)),k11_newton(k1_nat_1(B,np__2)))) ) ) ) ) )).

fof(t41_series_2,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)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_xcmplx_0(k3_xcmplx_0(A,D),B) )
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(C),D) = k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(A,k1_nat_1(D,np__1)),D),np__2),k3_xcmplx_0(D,B)),B) ) ) ) ) ) )).

fof(t42_series_2,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)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_xcmplx_0(k3_xcmplx_0(A,D),B) )
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(C),D) = k7_xcmplx_0(k3_xcmplx_0(k1_nat_1(D,np__1),k2_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,C,np__0),k2_seq_1(k5_numbers,k1_numbers,C,D))),np__2) ) ) ) ) ) )).
%------------------------------------------------------------------------------