
theorem Th16:
  for R being antisymmetric reflexive transitive with_suprema non
  empty RelStr, D being lower directed Subset of R for x, y being Element of R
  st x in D & y in D holds x "\/" y in D
proof
  let R be antisymmetric reflexive transitive with_suprema non empty RelStr,
  D be lower directed Subset of R;
  let x, y be Element of R;
  assume x in D & y in D;
  then consider z being Element of R such that
A1: z in D and
A2: x <= z & y <= z by WAYBEL_0:def 1;
  x "\/" y <= z by A2,YELLOW_0:22;
  hence thesis by A1,WAYBEL_0:def 19;
end;
