reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;

theorem Th1:
  for A,B be Matrix of K for nt be Element of n-tuples_on NAT, mt
be Element of m-tuples_on NAT st [:rng nt,rng mt:] c= Indices A holds Segm(A+B,
  nt,mt) = Segm(A,nt,mt) + Segm(B,nt,mt)
proof
  let A,B be Matrix of K;
  let nt be Element of n-tuples_on NAT, mt be Element of m-tuples_on NAT such
  that
A1: [:rng nt,rng mt:] c= Indices A;
  now
A2: Indices Segm(A,nt,mt) = Indices Segm(B,nt,mt) by MATRIX_0:26;
    let i,j such that
A3: [i,j] in Indices Segm(A+B,nt,mt);
    reconsider nti = nt.i, mtj = mt.j as Nat by VALUED_0:12;
A4: Indices Segm(A+B,nt,mt) = Indices Segm(A,nt,mt) by MATRIX_0:26;
    then
A5: [nt.i,mt.j] in Indices A by A1,A3,MATRIX13:17;
    thus Segm(A+B,nt,mt)*(i,j) = (A+B)*(nti,mtj) by A3,MATRIX13:def 1
      .= A*(nti,mtj)+B*(nti,mtj) by A5,MATRIX_3:def 3
      .= Segm(A,nt,mt)*(i,j) + B*(nti,mtj) by A3,A4,MATRIX13:def 1
      .= Segm(A,nt,mt)*(i,j) + Segm(B,nt,mt)*(i,j) by A3,A4,A2,MATRIX13:def 1
      .= (Segm(A,nt,mt) + Segm(B,nt,mt))*(i,j) by A3,A4,MATRIX_3:def 3;
  end;
  hence thesis by MATRIX_0:27;
end;
