
theorem
  for L being non empty transitive RelStr
  for S being directed-sups-inheriting non empty full SubRelStr of L
  for X being directed Subset of S st X <> {} & ex_sup_of X,L
  holds ex_sup_of X,S & "\/"(X,S) = "\/"(X,L)
proof
  let L be non empty transitive RelStr;
  let S be directed-sups-inheriting non empty full SubRelStr of L;
  let X be directed Subset of S;
  assume that
A1: X <> {} and
A2: ex_sup_of X,L;
  "\/"(X,L) in the carrier of S by A1,A2,Def4;
  hence thesis by A2,YELLOW_0:64;
end;
