
theorem Th23:
  for S, T being up-complete non empty Poset, x being Element of
  [:S,T:] st x`1 is compact & x`2 is compact holds x is compact
proof
  let S, T be up-complete non empty Poset, x be Element of [:S,T:];
  assume x`1 << x`1 & x`2 << x`2;
  hence x << x by Th21;
end;
