SET007 Axioms: SET007+100.ax
%------------------------------------------------------------------------------
% File : SET007+100 : TPTP v9.0.0. 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 unt; 0 def)
% Number of atoms : 544 ( 134 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 370 ( 4 ~; 0 |; 48 &)
% ( 2 <=>; 316 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 49 ( 49 usr; 10 con; 0-6 aty)
% Number of variables : 335 ( 332 !; 3 ?)
% 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) ) ).
%------------------------------------------------------------------------------