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

theorem Th21:
  for M, N being symmetric non empty MetrStruct holds max-Prod2(
  M,N) is symmetric
proof
  let M, N be symmetric non empty MetrStruct;
  let a, b be Element of max-Prod2(M,N);
  consider x1, y1 being Point of M, x2, y2 being Point of N such that
A1: a = [x1,x2] and
A2: b = [y1,y2] and
A3: (the distance of max-Prod2(M,N)).(a,b) = max ((the distance of M).(
  x1,y1),(the distance of N).(x2,y2)) by Def1;
  consider m1, n1 being Point of M, m2, n2 being Point of N such that
A4: b = [m1,m2] and
A5: a = [n1,n2] and
A6: (the distance of max-Prod2(M,N)).(b,a) = max ((the distance of M).(
  m1,n1),(the distance of N).(m2,n2)) by Def1;
A7: x1 = n1 by A1,A5,XTUPLE_0:1;
  the distance of N is symmetric by METRIC_1:def 8;
  then
A8: (the distance of N).(x2,y2) = (the distance of N).(y2,x2);
  the distance of M is symmetric by METRIC_1:def 8;
  then
A9: (the distance of M).(x1,y1) = (the distance of M).(y1,x1);
  y1 = m1 & y2 = m2 by A2,A4,XTUPLE_0:1;
  hence thesis by A1,A3,A5,A6,A9,A8,A7,XTUPLE_0:1;
end;
