
theorem Th30:
  for L being antisymmetric RelStr for a being Element of L, X
being set holds a = "\/"(X,L) & ex_sup_of X,L iff a is_>=_than X & for b being
  Element of L st b is_>=_than X holds a <= b
proof
  let L be antisymmetric RelStr;
  let a be Element of L, X be set;
  (a is_>=_than X & for b being Element of L st b is_>=_than X holds a <=
  b ) implies ex_sup_of X,L by Th15;
  hence thesis by Def9;
end;
