reserve i, j, n for Element of NAT,
  f, g, h, k for FinSequence of REAL,
  M, N for non empty MetrSpace;

theorem
  for M, N being non empty MetrStruct, m, n being Point of max-Prod2(M,N
  ) holds dist(m,n) = max (dist(m`1,n`1),dist(m`2,n`2))
proof
  let M, N be non empty MetrStruct, m, n be Point of max-Prod2(M,N);
  consider x1, y1 being Point of M, x2, y2 being Point of N such that
A1: m = [x1,x2] and
A2: n = [y1,y2] and
A3: (the distance of max-Prod2(M,N)).(m,n) = max ((the distance of M).(
  x1,y1),(the distance of N).(x2,y2)) by Def1;
A4: m`2 = x2 by A1;
  m`1 = x1 & n`1 = y1 by A1,A2;
  hence thesis by A2,A3,A4;
end;
