SET007 Axioms: SET007+100.ax


%------------------------------------------------------------------------------
% File     : SET007+100 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Sum and Product of Finite Sequences of Real Numbers
% 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    : rvsum_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  178 (  36 unit)
%            Number of atoms       :  544 ( 134 equality)
%            Maximal formula depth :   14 (   5 average)
%            Number of connectives :  370 (   4 ~  ;   0  |;  48  &)
%                                         (   2 <=>; 316 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   18 (   1 propositional; 0-3 arity)
%            Number of functors    :   49 (  10 constant; 0-6 arity)
%            Number of variables   :  335 (   0 singleton; 332 !;   3 ?)
%            Maximal term depth    :    5 (   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_rvsum_1,axiom,(
    $true )).

fof(t2_rvsum_1,axiom,(
    $true )).

fof(t3_rvsum_1,axiom,(
    r3_binop_1(k1_numbers,np__0,k33_binop_2) )).

fof(d1_rvsum_1,axiom,(
    k34_binop_2 = k8_finseqop(k1_numbers,k33_binop_2,k6_partfun1(k1_numbers),k31_binop_2) )).

fof(d2_rvsum_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k1_numbers,k1_numbers)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( A = k1_rvsum_1
      <=> ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => k2_seq_1(k1_numbers,k1_numbers,A,B) = k7_square_1(B) ) ) ) )).

fof(t4_rvsum_1,axiom,(
    $true )).

fof(t5_rvsum_1,axiom,(
    $true )).

fof(t6_rvsum_1,axiom,(
    $true )).

fof(t7_rvsum_1,axiom,(
    $true )).

fof(t8_rvsum_1,axiom,(
    $true )).

fof(t9_rvsum_1,axiom,(
    $true )).

fof(t10_rvsum_1,axiom,(
    $true )).

fof(t11_rvsum_1,axiom,(
    $true )).

fof(t12_rvsum_1,axiom,(
    $true )).

fof(t13_rvsum_1,axiom,(
    r3_binop_1(k1_numbers,np__1,k35_binop_2) )).

fof(t14_rvsum_1,axiom,(
    $true )).

fof(t15_rvsum_1,axiom,(
    $true )).

fof(t16_rvsum_1,axiom,(
    r6_binop_1(k1_numbers,k35_binop_2,k33_binop_2) )).

fof(t17_rvsum_1,axiom,(
    r7_binop_1(k1_numbers,k1_rvsum_1,k35_binop_2) )).

fof(d3_rvsum_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k2_rvsum_1(A) = k5_funcop_1(k35_binop_2,A,k6_partfun1(k1_numbers)) ) )).

fof(t18_rvsum_1,axiom,(
    $true )).

fof(t19_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k2_seq_1(k1_numbers,k1_numbers,k2_rvsum_1(A),B) = k11_binop_2(A,B) ) ) )).

fof(t20_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => r7_binop_1(k1_numbers,k2_rvsum_1(A),k33_binop_2) ) )).

fof(t21_rvsum_1,axiom,(
    r1_finseqop(k1_numbers,k31_binop_2,k33_binop_2) )).

fof(t22_rvsum_1,axiom,(
    v1_finseqop(k33_binop_2,k1_numbers) )).

fof(t23_rvsum_1,axiom,(
    k6_finseqop(k1_numbers,k33_binop_2) = k31_binop_2 )).

fof(t24_rvsum_1,axiom,(
    r7_binop_1(k1_numbers,k31_binop_2,k33_binop_2) )).

fof(d4_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k3_rvsum_1(A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,A,B) ) ) )).

fof(t25_rvsum_1,axiom,(
    $true )).

fof(t26_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => ( r2_hidden(A,k4_finseq_1(k3_rvsum_1(B,C)))
               => k2_seq_1(k5_numbers,k1_numbers,k3_rvsum_1(B,C),A) = k9_binop_2(k2_seq_1(k5_numbers,k1_numbers,B,A),k2_seq_1(k5_numbers,k1_numbers,C,A)) ) ) ) ) )).

fof(t27_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k2_seq_1(k5_numbers,k1_numbers,k4_rvsum_1(A,C,D),B) = k9_binop_2(k2_seq_1(k5_numbers,k1_numbers,C,B),k2_seq_1(k5_numbers,k1_numbers,D,B)) ) ) ) ) )).

fof(t28_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k3_rvsum_1(k6_finseq_1(k1_numbers),A) = k6_finseq_1(k1_numbers) ) )).

fof(t29_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k3_rvsum_1(k12_finseq_1(k1_numbers,A),k12_finseq_1(k1_numbers,B)) = k12_finseq_1(k1_numbers,k9_binop_2(A,B)) ) ) )).

fof(t30_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k4_rvsum_1(A,k4_finseqop(k1_numbers,A,B),k4_finseqop(k1_numbers,A,C)) = k4_finseqop(k1_numbers,A,k9_binop_2(B,C)) ) ) ) )).

fof(t31_rvsum_1,axiom,(
    $true )).

fof(t32_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k4_rvsum_1(A,B,k4_rvsum_1(A,C,D)) = k4_rvsum_1(A,k4_rvsum_1(A,B,C),D) ) ) ) ) )).

fof(t33_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k4_rvsum_1(A,B,k4_finseqop(k1_numbers,A,np__0)) = B ) ) )).

fof(d5_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k5_rvsum_1(A) = k5_finseqop(k1_numbers,k1_numbers,A,k31_binop_2) ) )).

fof(t34_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k4_finseq_1(A) = k4_finseq_1(k5_rvsum_1(A)) ) )).

fof(t35_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k2_seq_1(k5_numbers,k1_numbers,k5_rvsum_1(B),A) = k7_binop_2(k2_seq_1(k5_numbers,k1_numbers,B,A)) ) ) )).

fof(t36_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => ( C = k2_seq_1(k5_numbers,k1_numbers,D,B)
                   => k2_seq_1(k5_numbers,k1_numbers,k6_rvsum_1(A,D),B) = k7_binop_2(C) ) ) ) ) ) )).

fof(t37_rvsum_1,axiom,(
    k5_rvsum_1(k6_finseq_1(k1_numbers)) = k6_finseq_1(k1_numbers) )).

fof(t38_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k5_rvsum_1(k12_finseq_1(k1_numbers,A)) = k12_finseq_1(k1_numbers,k7_binop_2(A)) ) )).

fof(t39_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k6_rvsum_1(A,k4_finseqop(k1_numbers,A,B)) = k4_finseqop(k1_numbers,A,k7_binop_2(B)) ) ) )).

fof(t40_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k4_rvsum_1(A,B,k6_rvsum_1(A,B)) = k4_finseqop(k1_numbers,A,np__0) ) ) )).

fof(t41_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ( k4_rvsum_1(A,B,C) = k4_finseqop(k1_numbers,A,np__0)
               => B = k6_rvsum_1(A,C) ) ) ) ) )).

fof(t42_rvsum_1,axiom,(
    $true )).

fof(t43_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ( k6_rvsum_1(A,B) = k6_rvsum_1(A,C)
               => B = C ) ) ) ) )).

fof(t44_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => ( k4_rvsum_1(A,B,C) = k4_rvsum_1(A,D,C)
                   => B = D ) ) ) ) ) )).

fof(t45_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k6_rvsum_1(A,k4_rvsum_1(A,B,C)) = k4_rvsum_1(A,k6_rvsum_1(A,B),k6_rvsum_1(A,C)) ) ) ) )).

fof(d6_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k7_rvsum_1(A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k34_binop_2,A,B) ) ) )).

fof(t46_rvsum_1,axiom,(
    $true )).

fof(t47_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => ( r2_hidden(A,k4_finseq_1(k7_rvsum_1(B,C)))
               => k2_seq_1(k5_numbers,k1_numbers,k7_rvsum_1(B,C),A) = k10_binop_2(k2_seq_1(k5_numbers,k1_numbers,B,A),k2_seq_1(k5_numbers,k1_numbers,C,A)) ) ) ) ) )).

fof(t48_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k2_seq_1(k5_numbers,k1_numbers,k8_rvsum_1(A,C,D),B) = k10_binop_2(k2_seq_1(k5_numbers,k1_numbers,C,B),k2_seq_1(k5_numbers,k1_numbers,D,B)) ) ) ) ) )).

fof(t49_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( k7_rvsum_1(k6_finseq_1(k1_numbers),A) = k6_finseq_1(k1_numbers)
        & k7_rvsum_1(A,k6_finseq_1(k1_numbers)) = k6_finseq_1(k1_numbers) ) ) )).

fof(t50_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k7_rvsum_1(k12_finseq_1(k1_numbers,A),k12_finseq_1(k1_numbers,B)) = k12_finseq_1(k1_numbers,k10_binop_2(A,B)) ) ) )).

fof(t51_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k8_rvsum_1(A,k4_finseqop(k1_numbers,A,B),k4_finseqop(k1_numbers,A,C)) = k4_finseqop(k1_numbers,A,k10_binop_2(B,C)) ) ) ) )).

fof(t52_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k8_rvsum_1(A,B,C) = k4_rvsum_1(A,B,k6_rvsum_1(A,C)) ) ) ) )).

fof(t53_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k8_rvsum_1(A,B,k4_finseqop(k1_numbers,A,np__0)) = B ) ) )).

fof(t54_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k8_rvsum_1(A,k4_finseqop(k1_numbers,A,np__0),B) = k6_rvsum_1(A,B) ) ) )).

fof(t55_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k8_rvsum_1(A,B,k6_rvsum_1(A,C)) = k4_rvsum_1(A,B,C) ) ) ) )).

fof(t56_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k6_rvsum_1(A,k8_rvsum_1(A,B,C)) = k8_rvsum_1(A,C,B) ) ) ) )).

fof(t57_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k6_rvsum_1(A,k8_rvsum_1(A,B,C)) = k4_rvsum_1(A,k6_rvsum_1(A,B),C) ) ) ) )).

fof(t58_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k8_rvsum_1(A,B,B) = k4_finseqop(k1_numbers,A,np__0) ) ) )).

fof(t59_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ( k8_rvsum_1(A,B,C) = k4_finseqop(k1_numbers,A,np__0)
               => B = C ) ) ) ) )).

fof(t60_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k8_rvsum_1(A,k8_rvsum_1(A,B,C),D) = k8_rvsum_1(A,B,k4_rvsum_1(A,C,D)) ) ) ) ) )).

fof(t61_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k4_rvsum_1(A,B,k8_rvsum_1(A,C,D)) = k8_rvsum_1(A,k4_rvsum_1(A,B,C),D) ) ) ) ) )).

fof(t62_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k8_rvsum_1(A,B,k8_rvsum_1(A,C,D)) = k4_rvsum_1(A,k8_rvsum_1(A,B,C),D) ) ) ) ) )).

fof(t63_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => B = k8_rvsum_1(A,k4_rvsum_1(A,B,C),C) ) ) ) )).

fof(t64_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => B = k4_rvsum_1(A,k8_rvsum_1(A,B,C),C) ) ) ) )).

fof(d7_rvsum_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k9_rvsum_1(A,B) = k5_finseqop(k1_numbers,k1_numbers,B,k2_rvsum_1(A)) ) ) )).

fof(t65_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k4_finseq_1(k9_rvsum_1(A,B)) = k4_finseq_1(B) ) ) )).

fof(t66_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => k2_seq_1(k5_numbers,k1_numbers,k9_rvsum_1(B,C),A) = k11_binop_2(B,k2_seq_1(k5_numbers,k1_numbers,C,A)) ) ) ) )).

fof(t67_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k2_seq_1(k5_numbers,k1_numbers,k10_rvsum_1(A,C,D),B) = k11_binop_2(C,k2_seq_1(k5_numbers,k1_numbers,D,B)) ) ) ) ) )).

fof(t68_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k9_rvsum_1(A,k6_finseq_1(k1_numbers)) = k6_finseq_1(k1_numbers) ) )).

fof(t69_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k9_rvsum_1(A,k12_finseq_1(k1_numbers,B)) = k12_finseq_1(k1_numbers,k11_binop_2(A,B)) ) ) )).

fof(t70_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k10_rvsum_1(A,B,k4_finseqop(k1_numbers,A,C)) = k4_finseqop(k1_numbers,A,k11_binop_2(B,C)) ) ) ) )).

fof(t71_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k10_rvsum_1(A,k11_binop_2(B,C),D) = k10_rvsum_1(A,B,k10_rvsum_1(A,C,D)) ) ) ) ) )).

fof(t72_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k10_rvsum_1(A,k9_binop_2(B,C),D) = k4_rvsum_1(A,k10_rvsum_1(A,B,D),k10_rvsum_1(A,C,D)) ) ) ) ) )).

fof(t73_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k10_rvsum_1(A,B,k4_rvsum_1(A,C,D)) = k4_rvsum_1(A,k10_rvsum_1(A,B,C),k10_rvsum_1(A,B,D)) ) ) ) ) )).

fof(t74_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k10_rvsum_1(A,np__1,B) = B ) ) )).

fof(t75_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k10_rvsum_1(A,np__0,B) = k4_finseqop(k1_numbers,A,np__0) ) ) )).

fof(t76_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k10_rvsum_1(A,k7_binop_2(np__1),B) = k6_rvsum_1(A,B) ) ) )).

fof(d8_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k11_rvsum_1(A) = k5_finseqop(k1_numbers,k1_numbers,A,k1_rvsum_1) ) )).

fof(t77_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k4_finseq_1(k11_rvsum_1(A)) = k4_finseq_1(A) ) )).

fof(t78_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k2_seq_1(k5_numbers,k1_numbers,k11_rvsum_1(B),A) = k7_square_1(k2_seq_1(k5_numbers,k1_numbers,B,A)) ) ) )).

fof(t79_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k2_seq_1(k5_numbers,k1_numbers,k12_rvsum_1(A,C),B) = k7_square_1(k2_seq_1(k5_numbers,k1_numbers,C,B)) ) ) ) )).

fof(t80_rvsum_1,axiom,(
    k11_rvsum_1(k6_finseq_1(k1_numbers)) = k6_finseq_1(k1_numbers) )).

fof(t81_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k11_rvsum_1(k12_finseq_1(k1_numbers,A)) = k12_finseq_1(k1_numbers,k7_square_1(A)) ) )).

fof(t82_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k12_rvsum_1(A,k4_finseqop(k1_numbers,A,B)) = k4_finseqop(k1_numbers,A,k7_square_1(B)) ) ) )).

fof(t83_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k12_rvsum_1(A,k6_rvsum_1(A,B)) = k12_rvsum_1(A,B) ) ) )).

fof(t84_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k12_rvsum_1(A,k10_rvsum_1(A,B,C)) = k10_rvsum_1(A,k7_square_1(B),k12_rvsum_1(A,C)) ) ) ) )).

fof(d9_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k13_rvsum_1(A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k35_binop_2,A,B) ) ) )).

fof(t85_rvsum_1,axiom,(
    $true )).

fof(t86_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => ( r2_hidden(A,k4_finseq_1(k13_rvsum_1(B,C)))
               => k2_seq_1(k5_numbers,k1_numbers,k13_rvsum_1(B,C),A) = k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,B,A),k2_seq_1(k5_numbers,k1_numbers,C,A)) ) ) ) ) )).

fof(t87_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k2_seq_1(k5_numbers,k1_numbers,k14_rvsum_1(A,C,D),B) = k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,C,B),k2_seq_1(k5_numbers,k1_numbers,D,B)) ) ) ) ) )).

fof(t88_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k13_rvsum_1(k6_finseq_1(k1_numbers),A) = k6_finseq_1(k1_numbers) ) )).

fof(t89_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k13_rvsum_1(k12_finseq_1(k1_numbers,A),k12_finseq_1(k1_numbers,B)) = k12_finseq_1(k1_numbers,k11_binop_2(A,B)) ) ) )).

fof(t90_rvsum_1,axiom,(
    $true )).

fof(t91_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k14_rvsum_1(A,B,k14_rvsum_1(A,C,D)) = k14_rvsum_1(A,k14_rvsum_1(A,B,C),D) ) ) ) ) )).

fof(t92_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k14_rvsum_1(A,k4_finseqop(k1_numbers,A,B),C) = k10_rvsum_1(A,B,C) ) ) ) )).

fof(t93_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k14_rvsum_1(A,k4_finseqop(k1_numbers,A,B),k4_finseqop(k1_numbers,A,C)) = k4_finseqop(k1_numbers,A,k11_binop_2(B,C)) ) ) ) )).

fof(t94_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers))
                 => k10_rvsum_1(A,B,k14_rvsum_1(A,C,D)) = k14_rvsum_1(A,k10_rvsum_1(A,B,C),D) ) ) ) ) )).

fof(t95_rvsum_1,axiom,(
    $true )).

fof(t96_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k10_rvsum_1(A,B,C) = k14_rvsum_1(A,k4_finseqop(k1_numbers,A,B),C) ) ) ) )).

fof(t97_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k12_rvsum_1(A,B) = k14_rvsum_1(A,B,B) ) ) )).

fof(t98_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k12_rvsum_1(A,k4_rvsum_1(A,B,C)) = k4_rvsum_1(A,k4_rvsum_1(A,k12_rvsum_1(A,B),k10_rvsum_1(A,np__2,k14_rvsum_1(A,B,C))),k12_rvsum_1(A,C)) ) ) ) )).

fof(t99_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k12_rvsum_1(A,k8_rvsum_1(A,B,C)) = k4_rvsum_1(A,k8_rvsum_1(A,k12_rvsum_1(A,B),k10_rvsum_1(A,np__2,k14_rvsum_1(A,B,C))),k12_rvsum_1(A,C)) ) ) ) )).

fof(t100_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k12_rvsum_1(A,k14_rvsum_1(A,B,C)) = k14_rvsum_1(A,k12_rvsum_1(A,B),k12_rvsum_1(A,C)) ) ) ) )).

fof(d10_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k15_rvsum_1(A) = k2_finsop_1(k1_numbers,A,k33_binop_2) ) )).

fof(t101_rvsum_1,axiom,(
    $true )).

fof(t102_rvsum_1,axiom,(
    k15_rvsum_1(k6_finseq_1(k1_numbers)) = np__0 )).

fof(t103_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k15_rvsum_1(k12_finseq_1(k1_numbers,A)) = A ) )).

fof(t104_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k15_rvsum_1(k8_finseq_1(k1_numbers,B,k12_finseq_1(k1_numbers,A))) = k9_binop_2(k15_rvsum_1(B),A) ) ) )).

fof(t105_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k15_rvsum_1(k8_finseq_1(k1_numbers,A,B)) = k9_binop_2(k15_rvsum_1(A),k15_rvsum_1(B)) ) ) )).

fof(t106_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k15_rvsum_1(k8_finseq_1(k1_numbers,k12_finseq_1(k1_numbers,A),B)) = k9_binop_2(A,k15_rvsum_1(B)) ) ) )).

fof(t107_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k15_rvsum_1(k2_finseq_4(k1_numbers,A,B)) = k9_binop_2(A,B) ) ) )).

fof(t108_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k15_rvsum_1(k3_finseq_4(k1_numbers,A,B,C)) = k9_binop_2(k9_binop_2(A,B),C) ) ) ) )).

fof(t109_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_2(A,k1_numbers,k4_finseq_2(np__0,k1_numbers))
     => k15_rvsum_1(A) = np__0 ) )).

fof(t110_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k15_rvsum_1(k4_finseqop(k1_numbers,A,B)) = k11_binop_2(A,B) ) ) )).

fof(t111_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k15_rvsum_1(k4_finseqop(k1_numbers,A,np__0)) = np__0 ) )).

fof(t112_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r2_hidden(D,k2_finseq_1(A))
                     => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) )
               => r1_xreal_0(k15_rvsum_1(B),k15_rvsum_1(C)) ) ) ) ) )).

fof(t113_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => ~ ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k2_finseq_1(A))
                       => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) )
                  & ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r2_hidden(D,k2_finseq_1(A))
                      & ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) )
                  & r1_xreal_0(k15_rvsum_1(C),k15_rvsum_1(B)) ) ) ) ) )).

fof(t114_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,k4_finseq_1(A))
             => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) )
       => r1_xreal_0(np__0,k15_rvsum_1(A)) ) ) )).

fof(t115_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ~ ( ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( r2_hidden(B,k4_finseq_1(A))
               => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) )
          & ? [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
              & r2_hidden(B,k4_finseq_1(A))
              & ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),np__0) )
          & r1_xreal_0(k15_rvsum_1(A),np__0) ) ) )).

fof(t116_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => r1_xreal_0(np__0,k15_rvsum_1(k11_rvsum_1(A))) ) )).

fof(t117_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k15_rvsum_1(k9_rvsum_1(A,B)) = k11_binop_2(A,k15_rvsum_1(B)) ) ) )).

fof(t118_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k15_rvsum_1(k5_rvsum_1(A)) = k7_binop_2(k15_rvsum_1(A)) ) )).

fof(t119_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k15_rvsum_1(k4_rvsum_1(A,B,C)) = k9_binop_2(k15_rvsum_1(B),k15_rvsum_1(C)) ) ) ) )).

fof(t120_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k15_rvsum_1(k8_rvsum_1(A,B,C)) = k10_binop_2(k15_rvsum_1(B),k15_rvsum_1(C)) ) ) ) )).

fof(t121_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ( k15_rvsum_1(k12_rvsum_1(A,B)) = np__0
           => B = k4_finseqop(k1_numbers,A,np__0) ) ) ) )).

fof(t122_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => r1_xreal_0(k7_square_1(k15_rvsum_1(k14_rvsum_1(A,B,C))),k11_binop_2(k15_rvsum_1(k12_rvsum_1(A,B)),k15_rvsum_1(k12_rvsum_1(A,C)))) ) ) ) )).

fof(d11_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k16_rvsum_1(A) = k2_finsop_1(k1_numbers,A,k35_binop_2) ) )).

fof(t123_rvsum_1,axiom,(
    $true )).

fof(t124_rvsum_1,axiom,(
    k16_rvsum_1(k6_finseq_1(k1_numbers)) = np__1 )).

fof(t125_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k16_rvsum_1(k12_finseq_1(k1_numbers,A)) = A ) )).

fof(t126_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k16_rvsum_1(k8_finseq_1(k1_numbers,B,k12_finseq_1(k1_numbers,A))) = k11_binop_2(k16_rvsum_1(B),A) ) ) )).

fof(t127_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k16_rvsum_1(k8_finseq_1(k1_numbers,A,B)) = k11_binop_2(k16_rvsum_1(A),k16_rvsum_1(B)) ) ) )).

fof(t128_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k16_rvsum_1(k8_finseq_1(k1_numbers,k12_finseq_1(k1_numbers,A),B)) = k11_binop_2(A,k16_rvsum_1(B)) ) ) )).

fof(t129_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k16_rvsum_1(k2_finseq_4(k1_numbers,A,B)) = k11_binop_2(A,B) ) ) )).

fof(t130_rvsum_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k16_rvsum_1(k3_finseq_4(k1_numbers,A,B,C)) = k11_binop_2(k11_binop_2(A,B),C) ) ) ) )).

fof(t131_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_2(A,k1_numbers,k4_finseq_2(np__0,k1_numbers))
     => k16_rvsum_1(A) = np__1 ) )).

fof(t132_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k16_rvsum_1(k4_finseqop(k1_numbers,A,np__1)) = np__1 ) )).

fof(t133_rvsum_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & r2_hidden(B,k4_finseq_1(A))
            & k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 )
      <=> k16_rvsum_1(A) = np__0 ) ) )).

fof(t134_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k16_rvsum_1(k4_finseqop(k1_numbers,k23_binop_2(A,B),C)) = k11_binop_2(k16_rvsum_1(k4_finseqop(k1_numbers,A,C)),k16_rvsum_1(k4_finseqop(k1_numbers,B,C))) ) ) ) )).

fof(t135_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k16_rvsum_1(k4_finseqop(k1_numbers,k24_binop_2(A,B),C)) = k16_rvsum_1(k4_finseqop(k1_numbers,B,k16_rvsum_1(k4_finseqop(k1_numbers,A,C)))) ) ) ) )).

fof(t136_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k16_rvsum_1(k4_finseqop(k1_numbers,A,k11_binop_2(B,C))) = k11_binop_2(k16_rvsum_1(k4_finseqop(k1_numbers,A,B)),k16_rvsum_1(k4_finseqop(k1_numbers,A,C))) ) ) ) )).

fof(t137_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k16_rvsum_1(k14_rvsum_1(A,B,C)) = k11_binop_2(k16_rvsum_1(B),k16_rvsum_1(C)) ) ) ) )).

fof(t138_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k16_rvsum_1(k10_rvsum_1(A,B,C)) = k11_binop_2(k16_rvsum_1(k4_finseqop(k1_numbers,A,B)),k16_rvsum_1(C)) ) ) ) )).

fof(t139_rvsum_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k16_rvsum_1(k12_rvsum_1(A,B)) = k7_square_1(k16_rvsum_1(B)) ) ) )).

fof(dt_k1_rvsum_1,axiom,
    ( v1_funct_1(k1_rvsum_1)
    & v1_funct_2(k1_rvsum_1,k1_numbers,k1_numbers)
    & m2_relset_1(k1_rvsum_1,k1_numbers,k1_numbers) )).

fof(dt_k2_rvsum_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_funct_1(k2_rvsum_1(A))
        & v1_funct_2(k2_rvsum_1(A),k1_numbers,k1_numbers)
        & m2_relset_1(k2_rvsum_1(A),k1_numbers,k1_numbers) ) ) )).

fof(dt_k3_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_finseq_1(B,k1_numbers) )
     => m2_finseq_1(k3_rvsum_1(A,B),k1_numbers) ) )).

fof(commutativity_k3_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_finseq_1(B,k1_numbers) )
     => k3_rvsum_1(A,B) = k3_rvsum_1(B,A) ) )).

fof(dt_k4_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => m2_finseq_2(k4_rvsum_1(A,B,C),k1_numbers,k4_finseq_2(A,k1_numbers)) ) )).

fof(commutativity_k4_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => k4_rvsum_1(A,B,C) = k4_rvsum_1(A,C,B) ) )).

fof(redefinition_k4_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => k4_rvsum_1(A,B,C) = k3_rvsum_1(B,C) ) )).

fof(dt_k5_rvsum_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m2_finseq_1(k5_rvsum_1(A),k1_numbers) ) )).

fof(involutiveness_k5_rvsum_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => k5_rvsum_1(k5_rvsum_1(A)) = A ) )).

fof(dt_k6_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers)) )
     => m2_finseq_2(k6_rvsum_1(A,B),k1_numbers,k4_finseq_2(A,k1_numbers)) ) )).

fof(involutiveness_k6_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers)) )
     => k6_rvsum_1(A,k6_rvsum_1(A,B)) = B ) )).

fof(redefinition_k6_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers)) )
     => k6_rvsum_1(A,B) = k5_rvsum_1(B) ) )).

fof(dt_k7_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_finseq_1(B,k1_numbers) )
     => m2_finseq_1(k7_rvsum_1(A,B),k1_numbers) ) )).

fof(dt_k8_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => m2_finseq_2(k8_rvsum_1(A,B,C),k1_numbers,k4_finseq_2(A,k1_numbers)) ) )).

fof(redefinition_k8_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => k8_rvsum_1(A,B,C) = k7_rvsum_1(B,C) ) )).

fof(dt_k9_rvsum_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & m1_finseq_1(B,k1_numbers) )
     => m2_finseq_1(k9_rvsum_1(A,B),k1_numbers) ) )).

fof(dt_k10_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_xreal_0(B)
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => m2_finseq_2(k10_rvsum_1(A,B,C),k1_numbers,k4_finseq_2(A,k1_numbers)) ) )).

fof(redefinition_k10_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_xreal_0(B)
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => k10_rvsum_1(A,B,C) = k9_rvsum_1(B,C) ) )).

fof(dt_k11_rvsum_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m2_finseq_1(k11_rvsum_1(A),k1_numbers) ) )).

fof(dt_k12_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers)) )
     => m2_finseq_2(k12_rvsum_1(A,B),k1_numbers,k4_finseq_2(A,k1_numbers)) ) )).

fof(redefinition_k12_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers)) )
     => k12_rvsum_1(A,B) = k11_rvsum_1(B) ) )).

fof(dt_k13_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_finseq_1(B,k1_numbers) )
     => m2_finseq_1(k13_rvsum_1(A,B),k1_numbers) ) )).

fof(commutativity_k13_rvsum_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_finseq_1(B,k1_numbers) )
     => k13_rvsum_1(A,B) = k13_rvsum_1(B,A) ) )).

fof(dt_k14_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => m2_finseq_2(k14_rvsum_1(A,B,C),k1_numbers,k4_finseq_2(A,k1_numbers)) ) )).

fof(commutativity_k14_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => k14_rvsum_1(A,B,C) = k14_rvsum_1(A,C,B) ) )).

fof(redefinition_k14_rvsum_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => k14_rvsum_1(A,B,C) = k13_rvsum_1(B,C) ) )).

fof(dt_k15_rvsum_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m1_subset_1(k15_rvsum_1(A),k1_numbers) ) )).

fof(dt_k16_rvsum_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m1_subset_1(k16_rvsum_1(A),k1_numbers) ) )).
%------------------------------------------------------------------------------