
theorem Th14:
  for L being antisymmetric non empty RelStr, X being Subset of [:
  L, L:] st X c= id the carrier of L & ex_sup_of X, [:L, L:] holds sup X in id
  the carrier of L
proof
  let L be antisymmetric non empty RelStr, X be Subset of [:L, L:];
  assume X c= id the carrier of L & ex_sup_of X, [:L, L:];
  then sup X = [sup proj1 X, sup proj2 X] & sup proj1 X = sup proj2 X by Th1
,Th8;
  hence thesis by RELAT_1:def 10;
end;
