reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;

theorem
  for M1,M2 being Matrix of n,m,D st (for i,j st [i,j] in Indices
  M1 holds M1*(i,j) = M2*(i,j)) holds M1 = M2
proof
  let M1,M2 be Matrix of n,m,D;
A1: len M1=n by Th25;
A2: len M2=n by Th25;
A3: now
    per cases;
    suppose
A4:   n =0;
      then width M1 = 0 by A1,Def3;
      hence width M1= width M2 by A2,A4,Def3;
    end;
    suppose
A5:   n > 0;
      then width M1= m by A1,Th20;
      hence width M1 = width M2 by A2,A5,Th20;
    end;
  end;
  assume for i,j st [i,j] in Indices M1 holds M1*(i,j) = M2*(i,j);
  hence thesis by A1,A2,A3,Th21;
end;
