
theorem Th22:
  for S, T being antisymmetric up-complete non empty reflexive
RelStr, x being Element of [:S,T:] st x is compact holds x`1 is compact & x`2
  is compact
proof
  let S, T be antisymmetric up-complete non empty reflexive RelStr, x be
  Element of [:S,T:];
  assume
A1: x << x;
  hence x`1 << x`1 by Th20;
  thus x`2 << x`2 by A1,Th20;
end;
