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

theorem Th18:
  for M, N being non empty MetrStruct, m1, m2 being Point of M, n1
, n2 being Point of N holds dist([m1,n1],[m2,n2]) = max (dist(m1,m2),dist(n1,n2
  ))
proof
  let M, N be non empty MetrStruct, m1, m2 be Point of M, n1, n2 be Point of N;
  consider x1, y1 being Point of M, x2, y2 being Point of N such that
A1: [m1,n1] = [x1,x2] and
A2: [m2,n2] = [y1,y2] and
A3: (the distance of max-Prod2(M,N)).([m1,n1],[m2,n2]) = max ((the
  distance of M).(x1,y1),(the distance of N).(x2,y2)) by Def1;
A4: m2 = y1 by A2,XTUPLE_0:1;
  m1 = x1 & n1 = x2 by A1,XTUPLE_0:1;
  hence thesis by A2,A3,A4,XTUPLE_0:1;
end;
