
theorem Th17:
  for R being up-complete non empty reflexive transitive antisymmetric
  TopRelStr, S being non empty directed Subset of R,
  a being Element of R holds a in S implies a <= "\/"(S, R)
proof
  let R be up-complete non empty reflexive transitive antisymmetric
  TopRelStr;
  let S be non empty directed Subset of R, a be Element of R;
  assume
A1: a in S;
  ex_sup_of S,R by WAYBEL_0:75;
  then S is_<=_than "\/"(S, R) by YELLOW_0:30;
  hence thesis by A1;
end;
