
theorem
  for L being non empty RelStr st for X being Subset of L holds
  ex_sup_of X,L holds L is complete
proof
  let L be non empty RelStr such that
A1: for X being Subset of L holds ex_sup_of X,L;
  let X be set;
  take "\/"(X,L);
  reconsider Y = X /\ the carrier of L as Subset of L by XBOOLE_1:17;
  ex_sup_of Y,L by A1;
  then ex_sup_of X,L by Th50;
  hence thesis by Def9;
end;
