reserve i,j for Nat;

theorem
  for D1,D2 being set, A being (Matrix of D1), B being Matrix of D2 st A
  =B holds for i,j st [i,j] in Indices A holds A*(i,j)=B*(i,j)
proof
  let D1,D2 be set, A be (Matrix of D1), B be Matrix of D2;
  assume
A1: A=B;
  let i,j;
  assume
A2: [i,j] in Indices A;
  then
A3: ex p being FinSequence of D1 st p= A.i & A*(i,j)=p.j by MATRIX_0:def 5;
  ex q being FinSequence of D2 st q= B.i & B*(i,j)=q.j by A1,A2,MATRIX_0:def 5;
  hence thesis by A1,A3;
end;
