SET007 Axioms: SET007+361.ax
%------------------------------------------------------------------------------
% File : SET007+361 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Sum and Product of Finite Sequences of Elements of a Field
% 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 : fvsum_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 145 ( 12 unt; 0 def)
% Number of atoms : 984 ( 136 equ)
% Maximal formula atoms : 17 ( 6 avg)
% Number of connectives : 974 ( 135 ~; 1 |; 434 &)
% ( 1 <=>; 403 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 34 ( 32 usr; 1 prp; 0-3 aty)
% Number of functors : 54 ( 54 usr; 4 con; 0-6 aty)
% Number of variables : 430 ( 429 !; 1 ?)
% 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(cc1_fvsum_1,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v5_rlvect_1(A) )
=> ( ~ v3_struct_0(A)
& v1_algstr_1(A) ) ) ) ).
fof(t1_fvsum_1,axiom,
$true ).
fof(t2_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A) )
=> v1_binop_1(u1_rlvect_1(A),u1_struct_0(A)) ) ).
fof(t3_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& l1_rlvect_1(A) )
=> v2_binop_1(u1_rlvect_1(A),u1_struct_0(A)) ) ).
fof(t4_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A) )
=> v1_binop_1(u1_group_1(A),u1_struct_0(A)) ) ).
fof(t5_fvsum_1,axiom,
$true ).
fof(t6_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> r3_binop_1(u1_struct_0(A),k2_group_1(A),u1_group_1(A)) ) ).
fof(t7_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k2_group_1(A) ) ).
fof(t8_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> r3_binop_1(u1_struct_0(A),k1_rlvect_1(A),u1_rlvect_1(A)) ) ).
fof(t9_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> k3_binop_1(u1_struct_0(A),u1_rlvect_1(A)) = k1_rlvect_1(A) ) ).
fof(t10_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) ) ).
fof(t11_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> v1_setwiseo(u1_group_1(A),u1_struct_0(A)) ) ).
fof(t12_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> r6_binop_1(u1_struct_0(A),u1_group_1(A),u1_rlvect_1(A)) ) ).
fof(d1_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_fvsum_1(A,B) = k8_funcop_1(u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B,k6_partfun1(u1_struct_0(A))) ) ) ).
fof(d2_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> k2_fvsum_1(A) = k8_finseqop(u1_struct_0(A),u1_rlvect_1(A),k6_partfun1(u1_struct_0(A)),k7_vectsp_1(A)) ) ).
fof(t13_fvsum_1,axiom,
$true ).
fof(t14_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k2_fvsum_1(A),B,C) = k6_rlvect_1(A,B,C) ) ) ) ).
fof(t15_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r7_binop_1(u1_struct_0(A),k1_fvsum_1(A,B),u1_rlvect_1(A)) ) ) ).
fof(t16_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> r1_finseqop(u1_struct_0(A),k7_vectsp_1(A),u1_rlvect_1(A)) ) ).
fof(t17_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> v1_finseqop(u1_rlvect_1(A),u1_struct_0(A)) ) ).
fof(t18_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> k6_finseqop(u1_struct_0(A),u1_rlvect_1(A)) = k7_vectsp_1(A) ) ).
fof(t19_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> r7_binop_1(u1_struct_0(A),k7_vectsp_1(A),u1_rlvect_1(A)) ) ).
fof(d3_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k3_fvsum_1(A,B,C) = k1_finseqop(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),B,C) ) ) ) ).
fof(t20_fvsum_1,axiom,
$true ).
fof(t21_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k3_setwop_2(k3_fvsum_1(A,B,C)))
& D = k1_funct_1(B,F)
& E = k1_funct_1(C,F) )
=> k1_funct_1(k3_fvsum_1(A,B,C),F) = k2_rlvect_1(A,D,E) ) ) ) ) ) ) ) ).
fof(t22_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_rlvect_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ! [F] :
( m2_finseq_2(F,u1_struct_0(C),k4_finseq_2(A,u1_struct_0(C)))
=> ! [G] :
( m2_finseq_2(G,u1_struct_0(C),k4_finseq_2(A,u1_struct_0(C)))
=> ( ( r2_hidden(B,k2_setwop_2(A))
& D = k1_funct_1(F,B)
& E = k1_funct_1(G,B) )
=> k1_funct_1(k4_fvsum_1(A,C,F,G),B) = k2_rlvect_1(C,D,E) ) ) ) ) ) ) ) ) ).
fof(t23_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( k3_fvsum_1(A,k6_finseq_1(u1_struct_0(A)),B) = k6_finseq_1(u1_struct_0(A))
& k3_fvsum_1(A,B,k6_finseq_1(u1_struct_0(A))) = k6_finseq_1(u1_struct_0(A)) ) ) ) ).
fof(t24_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_fvsum_1(np__1,A,k4_matrix_2(u1_struct_0(A),B),k4_matrix_2(u1_struct_0(A),C)) = k4_matrix_2(u1_struct_0(A),k2_rlvect_1(A,B,C)) ) ) ) ).
fof(t25_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k4_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,C),k4_finseqop(u1_struct_0(B),A,D)) = k4_finseqop(u1_struct_0(B),A,k2_rlvect_1(B,C,D)) ) ) ) ) ).
fof(t26_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k4_fvsum_1(A,B,C,D) = k4_fvsum_1(A,B,D,C) ) ) ) ) ).
fof(t27_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k4_fvsum_1(A,B,C,k4_fvsum_1(A,B,D,E)) = k4_fvsum_1(A,B,k4_fvsum_1(A,B,C,D),E) ) ) ) ) ) ).
fof(t28_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v5_rlvect_1(B)
& v1_algstr_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ( k4_fvsum_1(A,B,C,k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B))) = C
& C = k4_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B)),C) ) ) ) ) ).
fof(d4_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> k5_fvsum_1(A,B) = k5_finseqop(u1_struct_0(A),u1_struct_0(A),B,k7_vectsp_1(A)) ) ) ).
fof(t29_fvsum_1,axiom,
$true ).
fof(t30_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(B))
=> ( ( r2_hidden(A,k3_setwop_2(k5_fvsum_1(B,D)))
& C = k1_funct_1(D,A) )
=> k1_funct_1(k5_fvsum_1(B,D),A) = k5_rlvect_1(B,C) ) ) ) ) ) ).
fof(t31_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_rlvect_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(C),k4_finseq_2(B,u1_struct_0(C)))
=> ( ( r2_hidden(A,k2_setwop_2(B))
& D = k1_funct_1(E,A) )
=> k1_funct_1(k6_fvsum_1(B,C,E),A) = k5_rlvect_1(C,D) ) ) ) ) ) ) ).
fof(t32_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> k5_fvsum_1(A,k6_finseq_1(u1_struct_0(A))) = k6_finseq_1(u1_struct_0(A)) ) ).
fof(t33_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_fvsum_1(np__1,A,k4_matrix_2(u1_struct_0(A),B)) = k4_matrix_2(u1_struct_0(A),k5_rlvect_1(A,B)) ) ) ).
fof(t34_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k6_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,C)) = k4_finseqop(u1_struct_0(B),A,k5_rlvect_1(B,C)) ) ) ) ).
fof(t35_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ( k4_fvsum_1(A,B,C,k6_fvsum_1(A,B,C)) = k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B))
& k4_fvsum_1(A,B,k6_fvsum_1(A,B,C),C) = k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B)) ) ) ) ) ).
fof(t36_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v1_algstr_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ( k4_fvsum_1(A,B,C,D) = k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B))
=> ( C = k6_fvsum_1(A,B,D)
& D = k6_fvsum_1(A,B,C) ) ) ) ) ) ) ).
fof(t37_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v1_algstr_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k6_fvsum_1(A,B,k6_fvsum_1(A,B,C)) = C ) ) ) ).
fof(t38_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v1_algstr_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ( k6_fvsum_1(A,B,C) = k6_fvsum_1(A,B,D)
=> C = D ) ) ) ) ) ).
fof(t39_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ( ( k4_fvsum_1(A,B,D,C) = k4_fvsum_1(A,B,E,C)
| k4_fvsum_1(A,B,D,C) = k4_fvsum_1(A,B,C,E) )
=> D = E ) ) ) ) ) ) ).
fof(t40_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k6_fvsum_1(A,B,k4_fvsum_1(A,B,C,D)) = k4_fvsum_1(A,B,k6_fvsum_1(A,B,C),k6_fvsum_1(A,B,D)) ) ) ) ) ).
fof(d5_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k7_fvsum_1(A,B,C) = k1_finseqop(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k2_fvsum_1(A),B,C) ) ) ) ).
fof(t41_fvsum_1,axiom,
$true ).
fof(t42_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(B))
=> ( ( r2_hidden(A,k3_setwop_2(k7_fvsum_1(B,E,F)))
& C = k1_funct_1(E,A)
& D = k1_funct_1(F,A) )
=> k1_funct_1(k7_fvsum_1(B,E,F),A) = k6_rlvect_1(B,C,D) ) ) ) ) ) ) ) ).
fof(t43_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_rlvect_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ! [F] :
( m2_finseq_2(F,u1_struct_0(C),k4_finseq_2(B,u1_struct_0(C)))
=> ! [G] :
( m2_finseq_2(G,u1_struct_0(C),k4_finseq_2(B,u1_struct_0(C)))
=> ( ( r2_hidden(A,k2_setwop_2(B))
& D = k1_funct_1(F,A)
& E = k1_funct_1(G,A) )
=> k1_funct_1(k8_fvsum_1(B,C,F,G),A) = k6_rlvect_1(C,D,E) ) ) ) ) ) ) ) ) ).
fof(t44_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( k7_fvsum_1(A,k6_finseq_1(u1_struct_0(A)),B) = k6_finseq_1(u1_struct_0(A))
& k7_fvsum_1(A,B,k6_finseq_1(u1_struct_0(A))) = k6_finseq_1(u1_struct_0(A)) ) ) ) ).
fof(t45_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_fvsum_1(np__1,A,k4_matrix_2(u1_struct_0(A),B),k4_matrix_2(u1_struct_0(A),C)) = k4_matrix_2(u1_struct_0(A),k6_rlvect_1(A,B,C)) ) ) ) ).
fof(t46_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k8_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,C),k4_finseqop(u1_struct_0(B),A,D)) = k4_finseqop(u1_struct_0(B),A,k6_rlvect_1(B,C,D)) ) ) ) ) ).
fof(t47_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k8_fvsum_1(A,B,C,D) = k4_fvsum_1(A,B,C,k6_fvsum_1(A,B,D)) ) ) ) ) ).
fof(t48_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v1_algstr_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k8_fvsum_1(A,B,C,k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B))) = C ) ) ) ).
fof(t49_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v5_rlvect_1(B)
& v1_algstr_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k8_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B)),C) = k6_fvsum_1(A,B,C) ) ) ) ).
fof(t50_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v1_algstr_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k8_fvsum_1(A,B,C,k6_fvsum_1(A,B,D)) = k4_fvsum_1(A,B,C,D) ) ) ) ) ).
fof(t51_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k6_fvsum_1(A,B,k8_fvsum_1(A,B,C,D)) = k8_fvsum_1(A,B,D,C) ) ) ) ) ).
fof(t52_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k6_fvsum_1(A,B,k8_fvsum_1(A,B,C,D)) = k4_fvsum_1(A,B,k6_fvsum_1(A,B,C),D) ) ) ) ) ).
fof(t53_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k8_fvsum_1(A,B,C,C) = k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B)) ) ) ) ).
fof(t54_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ( k8_fvsum_1(A,B,C,D) = k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B))
=> C = D ) ) ) ) ) ).
fof(t55_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k8_fvsum_1(A,B,k8_fvsum_1(A,B,C,D),E) = k8_fvsum_1(A,B,C,k4_fvsum_1(A,B,D,E)) ) ) ) ) ) ).
fof(t56_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k4_fvsum_1(A,B,C,k8_fvsum_1(A,B,D,E)) = k8_fvsum_1(A,B,k4_fvsum_1(A,B,C,D),E) ) ) ) ) ) ).
fof(t57_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k8_fvsum_1(A,B,C,k8_fvsum_1(A,B,D,E)) = k4_fvsum_1(A,B,k8_fvsum_1(A,B,C,D),E) ) ) ) ) ) ).
fof(t58_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> C = k8_fvsum_1(A,B,k4_fvsum_1(A,B,C,D),D) ) ) ) ) ).
fof(t59_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> C = k4_fvsum_1(A,B,k8_fvsum_1(A,B,C,D),D) ) ) ) ) ).
fof(t60_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),k8_funcop_1(u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B,k6_partfun1(u1_struct_0(A))),C) = k1_group_1(A,B,C) ) ) ) ).
fof(t61_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),k1_fvsum_1(A,B),C) = k1_group_1(A,B,C) ) ) ) ).
fof(d6_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k9_fvsum_1(A,B,C) = k5_finseqop(u1_struct_0(A),u1_struct_0(A),B,k1_fvsum_1(A,C)) ) ) ) ).
fof(t62_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ( ( r2_hidden(A,k3_setwop_2(k9_fvsum_1(B,E,C)))
& D = k1_funct_1(E,A) )
=> k1_funct_1(k9_fvsum_1(B,E,C),A) = k1_group_1(B,C,D) ) ) ) ) ) ) ).
fof(t63_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ! [F] :
( m2_finseq_2(F,u1_struct_0(C),k4_finseq_2(B,u1_struct_0(C)))
=> ( ( r2_hidden(A,k2_setwop_2(B))
& D = k1_funct_1(F,A) )
=> k1_funct_1(k10_fvsum_1(B,C,F,E),A) = k1_group_1(C,E,D) ) ) ) ) ) ) ) ).
fof(t64_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k9_fvsum_1(A,k6_finseq_1(u1_struct_0(A)),B) = k6_finseq_1(u1_struct_0(A)) ) ) ).
fof(t65_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k10_fvsum_1(np__1,A,k4_matrix_2(u1_struct_0(A),C),B) = k4_matrix_2(u1_struct_0(A),k1_group_1(A,B,C)) ) ) ) ).
fof(t66_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k10_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,D),C) = k4_finseqop(u1_struct_0(B),A,k1_group_1(B,C,D)) ) ) ) ) ).
fof(t67_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k10_fvsum_1(A,B,E,k1_group_1(B,C,D)) = k10_fvsum_1(A,B,k10_fvsum_1(A,B,E,D),C) ) ) ) ) ) ).
fof(t68_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v7_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k10_fvsum_1(A,B,E,k2_rlvect_1(B,C,D)) = k4_fvsum_1(A,B,k10_fvsum_1(A,B,E,C),k10_fvsum_1(A,B,E,D)) ) ) ) ) ) ).
fof(t69_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v7_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k10_fvsum_1(A,B,k4_fvsum_1(A,B,D,E),C) = k4_fvsum_1(A,B,k10_fvsum_1(A,B,D,C),k10_fvsum_1(A,B,E,C)) ) ) ) ) ) ).
fof(t70_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k10_fvsum_1(A,B,C,k2_group_1(B)) = C ) ) ) ).
fof(t71_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k10_fvsum_1(A,B,C,k1_rlvect_1(B)) = k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B)) ) ) ) ).
fof(t72_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k10_fvsum_1(A,B,C,k5_rlvect_1(B,k2_group_1(B))) = k6_fvsum_1(A,B,C) ) ) ) ).
fof(d7_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k11_fvsum_1(A,B,C) = k1_finseqop(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B,C) ) ) ) ).
fof(t73_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(B))
=> ( ( r2_hidden(A,k3_setwop_2(k11_fvsum_1(B,E,F)))
& C = k1_funct_1(E,A)
& D = k1_funct_1(F,A) )
=> k1_funct_1(k11_fvsum_1(B,E,F),A) = k1_group_1(B,C,D) ) ) ) ) ) ) ) ).
fof(t74_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ! [F] :
( m2_finseq_2(F,u1_struct_0(C),k4_finseq_2(B,u1_struct_0(C)))
=> ! [G] :
( m2_finseq_2(G,u1_struct_0(C),k4_finseq_2(B,u1_struct_0(C)))
=> ( ( r2_hidden(A,k2_setwop_2(B))
& D = k1_funct_1(F,A)
& E = k1_funct_1(G,A) )
=> k1_funct_1(k12_fvsum_1(B,C,F,G),A) = k1_group_1(C,D,E) ) ) ) ) ) ) ) ) ).
fof(t75_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( k11_fvsum_1(A,k6_finseq_1(u1_struct_0(A)),B) = k6_finseq_1(u1_struct_0(A))
& k11_fvsum_1(A,B,k6_finseq_1(u1_struct_0(A))) = k6_finseq_1(u1_struct_0(A)) ) ) ) ).
fof(t76_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k12_fvsum_1(np__1,A,k4_matrix_2(u1_struct_0(A),B),k4_matrix_2(u1_struct_0(A),C)) = k4_matrix_2(u1_struct_0(A),k1_group_1(A,B,C)) ) ) ) ).
fof(t77_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v7_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k12_fvsum_1(A,B,C,D) = k12_fvsum_1(A,B,D,C) ) ) ) ) ).
fof(t78_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k11_fvsum_1(A,B,C) = k11_fvsum_1(A,C,B) ) ) ) ).
fof(t79_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k12_fvsum_1(A,B,C,k12_fvsum_1(A,B,D,E)) = k12_fvsum_1(A,B,k12_fvsum_1(A,B,C,D),E) ) ) ) ) ) ).
fof(t80_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ( k12_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,C),D) = k10_fvsum_1(A,B,D,C)
& k12_fvsum_1(A,B,D,k4_finseqop(u1_struct_0(B),A,C)) = k10_fvsum_1(A,B,D,C) ) ) ) ) ) ).
fof(t81_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k12_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,C),k4_finseqop(u1_struct_0(B),A,D)) = k4_finseqop(u1_struct_0(B),A,k10_group_1(B,C,D)) ) ) ) ) ).
fof(t82_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k10_fvsum_1(A,B,k12_fvsum_1(A,B,D,E),C) = k12_fvsum_1(A,B,k10_fvsum_1(A,B,D,C),E) ) ) ) ) ) ).
fof(t83_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ( k10_fvsum_1(A,B,k12_fvsum_1(A,B,D,E),C) = k12_fvsum_1(A,B,k10_fvsum_1(A,B,D,C),E)
& k10_fvsum_1(A,B,k12_fvsum_1(A,B,D,E),C) = k12_fvsum_1(A,B,D,k10_fvsum_1(A,B,E,C)) ) ) ) ) ) ) ).
fof(t84_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k10_fvsum_1(A,B,D,C) = k12_fvsum_1(A,B,k4_finseqop(u1_struct_0(B),A,C),D) ) ) ) ) ).
fof(d8_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> k9_rlvect_1(A,B) = k2_finsop_1(u1_struct_0(A),B,u1_rlvect_1(A)) ) ) ).
fof(t85_fvsum_1,axiom,
$true ).
fof(t86_fvsum_1,axiom,
$true ).
fof(t87_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k9_rlvect_1(A,k8_finseq_1(u1_struct_0(A),C,k4_matrix_2(u1_struct_0(A),B))) = k2_rlvect_1(A,k9_rlvect_1(A,C),B) ) ) ) ).
fof(t88_fvsum_1,axiom,
$true ).
fof(t89_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k9_rlvect_1(A,k8_finseq_1(u1_struct_0(A),k4_matrix_2(u1_struct_0(A),B),C)) = k2_rlvect_1(A,B,k9_rlvect_1(A,C)) ) ) ) ).
fof(t90_fvsum_1,axiom,
$true ).
fof(t91_fvsum_1,axiom,
$true ).
fof(t92_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k9_rlvect_1(A,k9_fvsum_1(A,C,B)) = k1_group_1(A,B,k9_rlvect_1(A,C)) ) ) ) ).
fof(t93_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k4_finseq_2(np__0,u1_struct_0(A)))
=> k9_rlvect_1(A,B) = k1_rlvect_1(A) ) ) ).
fof(t94_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> k9_rlvect_1(A,k5_fvsum_1(A,B)) = k5_rlvect_1(A,k9_rlvect_1(A,B)) ) ) ).
fof(t95_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k9_rlvect_1(B,k4_fvsum_1(A,B,C,D)) = k4_rlvect_1(B,k9_rlvect_1(B,C),k9_rlvect_1(B,D)) ) ) ) ) ).
fof(t96_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k9_rlvect_1(B,k8_fvsum_1(A,B,C,D)) = k6_rlvect_1(B,k9_rlvect_1(B,C),k9_rlvect_1(B,D)) ) ) ) ) ).
fof(d9_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> k13_fvsum_1(A,B) = k2_finsop_1(u1_struct_0(A),B,u1_group_1(A)) ) ) ).
fof(t97_fvsum_1,axiom,
$true ).
fof(t98_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> k13_fvsum_1(A,k6_finseq_1(u1_struct_0(A))) = k2_group_1(A) ) ).
fof(t99_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k13_fvsum_1(A,k4_matrix_2(u1_struct_0(A),B)) = B ) ) ).
fof(t100_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k13_fvsum_1(A,k8_finseq_1(u1_struct_0(A),C,k4_matrix_2(u1_struct_0(A),B))) = k10_group_1(A,k13_fvsum_1(A,C),B) ) ) ) ).
fof(t101_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k13_fvsum_1(A,k8_finseq_1(u1_struct_0(A),B,C)) = k10_group_1(A,k13_fvsum_1(A,B),k13_fvsum_1(A,C)) ) ) ) ).
fof(t102_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k13_fvsum_1(A,k8_finseq_1(u1_struct_0(A),k4_matrix_2(u1_struct_0(A),B),C)) = k10_group_1(A,B,k13_fvsum_1(A,C)) ) ) ) ).
fof(t103_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k13_fvsum_1(A,k7_rlvect_1(A,B,C)) = k10_group_1(A,B,C) ) ) ) ).
fof(t104_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k13_fvsum_1(A,k8_rlvect_1(A,B,C,D)) = k10_group_1(A,k10_group_1(A,B,C),D) ) ) ) ) ).
fof(t105_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k4_finseq_2(np__0,u1_struct_0(A)))
=> k13_fvsum_1(A,B) = k2_group_1(A) ) ) ).
fof(t106_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> k13_fvsum_1(B,k4_finseqop(u1_struct_0(B),A,k2_group_1(B))) = k2_group_1(B) ) ) ).
fof(t107_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r2_hidden(C,k3_setwop_2(B))
& k1_funct_1(B,C) = k1_rlvect_1(A) )
<=> k13_fvsum_1(A,B) = k1_rlvect_1(A) ) ) ) ).
fof(t108_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v4_group_1(C)
& v7_group_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k13_fvsum_1(C,k4_finseqop(u1_struct_0(C),k1_nat_1(A,B),D)) = k10_group_1(C,k13_fvsum_1(C,k4_finseqop(u1_struct_0(C),A,D)),k13_fvsum_1(C,k4_finseqop(u1_struct_0(C),B,D))) ) ) ) ) ).
fof(t109_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v4_group_1(C)
& v7_group_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> k13_fvsum_1(C,k4_finseqop(u1_struct_0(C),k2_nat_1(A,B),D)) = k13_fvsum_1(C,k4_finseqop(u1_struct_0(C),B,k13_fvsum_1(C,k4_finseqop(u1_struct_0(C),A,D)))) ) ) ) ) ).
fof(t110_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k13_fvsum_1(B,k4_finseqop(u1_struct_0(B),A,k10_group_1(B,C,D))) = k10_group_1(B,k13_fvsum_1(B,k4_finseqop(u1_struct_0(B),A,C)),k13_fvsum_1(B,k4_finseqop(u1_struct_0(B),A,D))) ) ) ) ) ).
fof(t111_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k13_fvsum_1(B,k12_fvsum_1(A,B,C,D)) = k10_group_1(B,k13_fvsum_1(B,C),k13_fvsum_1(B,D)) ) ) ) ) ).
fof(t112_fvsum_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B)))
=> k13_fvsum_1(B,k10_fvsum_1(A,B,D,C)) = k10_group_1(B,k13_fvsum_1(B,k4_finseqop(u1_struct_0(B),A,C)),k13_fvsum_1(B,D)) ) ) ) ) ).
fof(d10_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k14_fvsum_1(A,B,C) = k9_rlvect_1(A,k11_fvsum_1(A,B,C)) ) ) ) ).
fof(t113_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k14_fvsum_1(A,k4_matrix_2(u1_struct_0(A),B),k4_matrix_2(u1_struct_0(A),C)) = k10_group_1(A,B,C) ) ) ) ).
fof(t114_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k14_fvsum_1(A,k7_rlvect_1(A,B,C),k7_rlvect_1(A,D,E)) = k4_rlvect_1(A,k10_group_1(A,B,D),k10_group_1(A,C,E)) ) ) ) ) ) ).
fof(t115_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k14_fvsum_1(A,B,C) = k14_fvsum_1(A,C,B) ) ) ) ).
fof(dt_k1_fvsum_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_funct_1(k1_fvsum_1(A,B))
& v1_funct_2(k1_fvsum_1(A,B),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k1_fvsum_1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k2_fvsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v1_funct_1(k2_fvsum_1(A))
& v1_funct_2(k2_fvsum_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(k2_fvsum_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_k3_fvsum_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_finseq_1(B,u1_struct_0(A))
& m1_finseq_1(C,u1_struct_0(A)) )
=> m2_finseq_1(k3_fvsum_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k4_fvsum_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B)))
& m1_subset_1(D,k4_finseq_2(A,u1_struct_0(B))) )
=> m2_finseq_2(k4_fvsum_1(A,B,C,D),u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B))) ) ).
fof(redefinition_k4_fvsum_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B)))
& m1_subset_1(D,k4_finseq_2(A,u1_struct_0(B))) )
=> k4_fvsum_1(A,B,C,D) = k3_fvsum_1(B,C,D) ) ).
fof(dt_k5_fvsum_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_finseq_1(B,u1_struct_0(A)) )
=> m2_finseq_1(k5_fvsum_1(A,B),u1_struct_0(A)) ) ).
fof(dt_k6_fvsum_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B))) )
=> m2_finseq_2(k6_fvsum_1(A,B,C),u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B))) ) ).
fof(redefinition_k6_fvsum_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B))) )
=> k6_fvsum_1(A,B,C) = k5_fvsum_1(B,C) ) ).
fof(dt_k7_fvsum_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_finseq_1(B,u1_struct_0(A))
& m1_finseq_1(C,u1_struct_0(A)) )
=> m2_finseq_1(k7_fvsum_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k8_fvsum_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B)))
& m1_subset_1(D,k4_finseq_2(A,u1_struct_0(B))) )
=> m2_finseq_2(k8_fvsum_1(A,B,C,D),u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B))) ) ).
fof(redefinition_k8_fvsum_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B)))
& m1_subset_1(D,k4_finseq_2(A,u1_struct_0(B))) )
=> k8_fvsum_1(A,B,C,D) = k7_fvsum_1(B,C,D) ) ).
fof(dt_k9_fvsum_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_finseq_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m2_finseq_1(k9_fvsum_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k10_fvsum_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_group_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B)))
& m1_subset_1(D,u1_struct_0(B)) )
=> m2_finseq_2(k10_fvsum_1(A,B,C,D),u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B))) ) ).
fof(redefinition_k10_fvsum_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_group_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B)))
& m1_subset_1(D,u1_struct_0(B)) )
=> k10_fvsum_1(A,B,C,D) = k9_fvsum_1(B,C,D) ) ).
fof(dt_k11_fvsum_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_finseq_1(B,u1_struct_0(A))
& m1_finseq_1(C,u1_struct_0(A)) )
=> m2_finseq_1(k11_fvsum_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k12_fvsum_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_group_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B)))
& m1_subset_1(D,k4_finseq_2(A,u1_struct_0(B))) )
=> m2_finseq_2(k12_fvsum_1(A,B,C,D),u1_struct_0(B),k4_finseq_2(A,u1_struct_0(B))) ) ).
fof(redefinition_k12_fvsum_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& l1_group_1(B)
& m1_subset_1(C,k4_finseq_2(A,u1_struct_0(B)))
& m1_subset_1(D,k4_finseq_2(A,u1_struct_0(B))) )
=> k12_fvsum_1(A,B,C,D) = k11_fvsum_1(B,C,D) ) ).
fof(dt_k13_fvsum_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_finseq_1(B,u1_struct_0(A)) )
=> m1_subset_1(k13_fvsum_1(A,B),u1_struct_0(A)) ) ).
fof(dt_k14_fvsum_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_finseq_1(B,u1_struct_0(A))
& m1_finseq_1(C,u1_struct_0(A)) )
=> m1_subset_1(k14_fvsum_1(A,B,C),u1_struct_0(A)) ) ).
%------------------------------------------------------------------------------