
theorem
  for L being complete non empty Poset for S being non empty full
  SubRelStr of L for X being Subset of S st "\/"(X,L) in the carrier of S holds
  "\/"(X,S) = "\/"(X,L)
proof
  let L be complete non empty Poset;
  let S be non empty full SubRelStr of L;
  let X be Subset of S such that
A1: "\/"(X,L) in the carrier of S;
  ex_sup_of X,L by Th17;
  hence thesis by A1,Th64;
end;
