SET007 Axioms: SET007+881.ax
%------------------------------------------------------------------------------
% File : SET007+881 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Theory of Matrices of Complex Elements
% 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 : matrix_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 2 unt; 0 def)
% Number of atoms : 433 ( 111 equ)
% Maximal formula atoms : 21 ( 7 avg)
% Number of connectives : 397 ( 24 ~; 21 |; 210 &)
% ( 1 <=>; 141 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 20 ( 19 usr; 0 prp; 1-3 aty)
% Number of functors : 36 ( 36 usr; 8 con; 0-4 aty)
% Number of variables : 118 ( 118 !; 0 ?)
% 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_matrix_5,axiom,
np__1 = k2_group_1(k1_complfld) ).
fof(t2_matrix_5,axiom,
k1_rlvect_1(k1_complfld) = np__0 ).
fof(d1_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k1_matrix_5(A) = A ) ).
fof(d2_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
=> k2_matrix_5(A) = A ) ).
fof(t3_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( k1_matrix_5(A) = k1_matrix_5(B)
=> A = B ) ) ) ).
fof(t4_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k1_complfld))) )
=> ( k2_matrix_5(A) = k2_matrix_5(B)
=> A = B ) ) ) ).
fof(t5_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> A = k2_matrix_5(k1_matrix_5(A)) ) ).
fof(t6_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
=> A = k1_matrix_5(k2_matrix_5(A)) ) ).
fof(d3_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> k3_matrix_5(A,B) = k2_matrix_5(k3_matrix_3(k1_complfld,k1_matrix_5(A),k1_matrix_5(B))) ) ) ).
fof(d4_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k4_matrix_5(A) = k2_matrix_5(k2_matrix_3(k1_complfld,k1_matrix_5(A))) ) ).
fof(d5_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> k5_matrix_5(A,B) = k2_matrix_5(k1_matrix_4(k1_complfld,k1_matrix_5(A),k1_matrix_5(B))) ) ) ).
fof(d6_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> k6_matrix_5(A,B) = k2_matrix_5(k4_matrix_3(k1_complfld,k1_matrix_5(A),k1_matrix_5(B))) ) ) ).
fof(d7_matrix_5,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( C = k7_matrix_5(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_complfld))
=> ( D = A
=> C = k2_matrix_5(k6_matrix_3(k1_complfld,k1_matrix_5(B),D)) ) ) ) ) ) ) ).
fof(t7_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( k3_finseq_1(A) = k3_finseq_1(k1_matrix_5(A))
& k1_matrix_1(A) = k1_matrix_1(k1_matrix_5(A)) ) ) ).
fof(t8_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
=> ( k3_finseq_1(A) = k3_finseq_1(k2_matrix_5(A))
& k1_matrix_1(A) = k1_matrix_1(k2_matrix_5(A)) ) ) ).
fof(t9_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> k4_matrix_5(k4_matrix_5(A)) = A ) ) ).
fof(t10_matrix_5,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] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
=> k6_matrix_3(A,B,k2_group_1(A)) = B ) ) ).
fof(t11_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k7_matrix_5(np__1,A) = A ) ).
fof(t12_matrix_5,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] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
=> k6_matrix_3(A,k6_matrix_3(A,D,C),B) = k6_matrix_3(A,D,k10_group_1(A,B,C)) ) ) ) ) ).
fof(t13_matrix_5,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] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
=> k6_matrix_3(A,D,k4_rlvect_1(A,B,C)) = k3_matrix_3(A,k6_matrix_3(A,D,B),k6_matrix_3(A,D,C)) ) ) ) ) ).
fof(t14_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k3_matrix_5(A,A) = k7_matrix_5(np__2,A) ) ).
fof(t15_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k3_matrix_5(k3_matrix_5(A,A),A) = k7_matrix_5(np__3,A) ) ).
fof(d8_matrix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_matrix_5(A,B) = k2_matrix_5(k1_matrix_3(k1_complfld,A,B)) ) ) ).
fof(t16_matrix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_matrix_5(k8_matrix_5(A,B)) = k1_matrix_3(k1_complfld,A,B) ) ) ).
fof(t17_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> k3_matrix_5(A,k4_matrix_5(A)) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) ).
fof(t18_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> k5_matrix_5(A,A) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) ).
fof(t19_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C)
& k1_matrix_1(A) = k1_matrix_1(B)
& k1_matrix_1(B) = k1_matrix_1(C)
& k3_matrix_5(A,C) = k3_matrix_5(B,C) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| A = B ) ) ) ) ) ).
fof(t20_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k3_finseq_1(B),np__0)
=> k5_matrix_5(A,k4_matrix_5(B)) = k3_matrix_5(A,B) ) ) ) ).
fof(t21_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B)
& A = k3_matrix_5(A,B) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| B = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) ) ) ).
fof(t22_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B)
& k5_matrix_5(A,B) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| A = B ) ) ) ) ).
fof(t23_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B)
& k3_matrix_5(A,B) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| B = k4_matrix_5(A) ) ) ) ) ).
fof(t24_matrix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k4_matrix_5(k8_matrix_5(A,B)) = k8_matrix_5(A,B) ) ) ) ).
fof(t25_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B)
& k5_matrix_5(B,A) = B )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| A = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) ) ) ).
fof(t26_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| A = k5_matrix_5(A,k5_matrix_5(B,B)) ) ) ) ) ).
fof(t27_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| k4_matrix_5(k3_matrix_5(A,B)) = k3_matrix_5(k4_matrix_5(A),k4_matrix_5(B)) ) ) ) ) ).
fof(t28_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| k5_matrix_5(A,k5_matrix_5(A,B)) = B ) ) ) ) ).
fof(t29_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C)
& k1_matrix_1(A) = k1_matrix_1(B)
& k1_matrix_1(B) = k1_matrix_1(C)
& k5_matrix_5(A,C) = k5_matrix_5(B,C) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| A = B ) ) ) ) ) ).
fof(t30_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C)
& k1_matrix_1(A) = k1_matrix_1(B)
& k1_matrix_1(B) = k1_matrix_1(C)
& k5_matrix_5(C,A) = k5_matrix_5(C,B) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| A = B ) ) ) ) ) ).
fof(t31_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(B) = k3_finseq_1(C)
& k1_matrix_1(B) = k1_matrix_1(C)
& k1_matrix_1(A) = k3_finseq_1(B) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| r1_xreal_0(k3_finseq_1(B),np__0)
| k6_matrix_5(A,k3_matrix_5(B,C)) = k3_matrix_5(k6_matrix_5(A,B),k6_matrix_5(A,C)) ) ) ) ) ) ).
fof(t32_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(B) = k3_finseq_1(C)
& k1_matrix_1(B) = k1_matrix_1(C)
& k3_finseq_1(A) = k1_matrix_1(B) )
=> ( r1_xreal_0(k3_finseq_1(B),np__0)
| r1_xreal_0(k3_finseq_1(A),np__0)
| k6_matrix_5(k3_matrix_5(B,C),A) = k3_matrix_5(k6_matrix_5(B,A),k6_matrix_5(C,A)) ) ) ) ) ) ).
fof(t33_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B) )
=> k3_matrix_5(A,B) = k3_matrix_5(B,A) ) ) ) ).
fof(t34_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(A) = k3_finseq_1(C)
& k1_matrix_1(A) = k1_matrix_1(B)
& k1_matrix_1(A) = k1_matrix_1(C) )
=> k3_matrix_5(k3_matrix_5(A,B),C) = k3_matrix_5(A,k3_matrix_5(B,C)) ) ) ) ) ).
fof(t35_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> k3_matrix_5(A,k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A))) = A ) ) ).
fof(t36_matrix_5,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] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& k1_matrix_1(C) = k1_matrix_1(D) )
=> ( r1_xreal_0(k3_finseq_1(C),np__0)
| k6_matrix_3(A,k3_matrix_3(A,C,D),B) = k3_matrix_3(A,k6_matrix_3(A,C,B),k6_matrix_3(A,D,B)) ) ) ) ) ) ) ).
fof(t37_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ! [C] :
( v1_xcmplx_0(C)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| k7_matrix_5(C,k3_matrix_5(A,B)) = k3_matrix_5(k7_matrix_5(C,A),k7_matrix_5(C,B)) ) ) ) ) ) ).
fof(t38_matrix_5,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] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
=> ( k1_matrix_1(B) = k3_finseq_1(C)
=> ( r1_xreal_0(k3_finseq_1(B),np__0)
| r1_xreal_0(k3_finseq_1(C),np__0)
| k4_matrix_3(A,k1_matrix_3(A,k3_finseq_1(B),k1_matrix_1(B)),C) = k1_matrix_3(A,k3_finseq_1(B),k1_matrix_1(C)) ) ) ) ) ) ).
fof(t39_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( k1_matrix_1(A) = k3_finseq_1(B)
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| r1_xreal_0(k3_finseq_1(B),np__0)
| k6_matrix_5(k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)),B) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(B)) ) ) ) ) ).
fof(t40_matrix_5,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] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
=> ( ~ r1_xreal_0(k3_finseq_1(B),np__0)
=> k6_matrix_3(A,B,k1_rlvect_1(A)) = k1_matrix_3(A,k3_finseq_1(B),k1_matrix_1(B)) ) ) ) ).
fof(t41_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> k7_matrix_5(np__0,A) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) ).
fof(t42_matrix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
=> ~ ( ~ r1_xreal_0(k3_finseq_1(C),np__0)
& ~ r1_xreal_0(k3_finseq_1(D),np__0)
& k1_matrix_1(C) = k3_finseq_1(D)
& r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(C))
& r1_xreal_0(np__1,B)
& r1_xreal_0(B,k1_matrix_1(D))
& ! [E] :
( m2_finseq_1(E,k2_numbers)
=> ~ ( k3_finseq_1(E) = k3_finseq_1(D)
& k9_matrix_5(E,np__1) = k5_binop_2(k3_matrix_1(k2_numbers,C,A,np__1),k3_matrix_1(k2_numbers,D,np__1,B))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,F)
=> ( r1_xreal_0(k3_finseq_1(D),F)
| k9_matrix_5(E,k23_binop_2(F,np__1)) = k3_binop_2(k9_matrix_5(E,F),k5_binop_2(k3_matrix_1(k2_numbers,C,A,k23_binop_2(F,np__1)),k3_matrix_1(k2_numbers,D,k23_binop_2(F,np__1),B))) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k1_matrix_5(A))
& m2_finseq_1(k1_matrix_5(A),k3_finseq_2(u1_struct_0(k1_complfld))) ) ) ).
fof(dt_k2_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
=> ( v1_matrix_1(k2_matrix_5(A))
& m2_finseq_1(k2_matrix_5(A),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k3_matrix_5,axiom,
! [A,B] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers))
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k3_matrix_5(A,B))
& m2_finseq_1(k3_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k4_matrix_5,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k4_matrix_5(A))
& m2_finseq_1(k4_matrix_5(A),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k5_matrix_5,axiom,
! [A,B] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers))
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k5_matrix_5(A,B))
& m2_finseq_1(k5_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k6_matrix_5,axiom,
! [A,B] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers))
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k6_matrix_5(A,B))
& m2_finseq_1(k6_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k7_matrix_5,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k7_matrix_5(A,B))
& m2_finseq_1(k7_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k8_matrix_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_matrix_1(k8_matrix_5(A,B))
& m2_finseq_1(k8_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k9_matrix_5,axiom,
! [A,B] :
( m1_finseq_1(A,k2_numbers)
=> m1_subset_1(k9_matrix_5(A,B),k2_numbers) ) ).
fof(redefinition_k9_matrix_5,axiom,
! [A,B] :
( m1_finseq_1(A,k2_numbers)
=> k9_matrix_5(A,B) = k1_funct_1(A,B) ) ).
%------------------------------------------------------------------------------