
theorem
  for R, S being lower-bounded antisymmetric reflexive transitive
with_suprema non empty RelStr st (the carrier of R) /\ (the carrier of S) is
  non empty lower directed Subset of S holds Bottom S in the carrier of R
proof
  let R, S be lower-bounded antisymmetric reflexive transitive with_suprema
  non empty RelStr;
  assume
A1: (the carrier of R) /\ (the carrier of S) is non empty lower directed
  Subset of S;
  then consider x being object such that
A2: x in (the carrier of R) /\ (the carrier of S) by XBOOLE_0:def 1;
  reconsider x as Element of S by A2,Th13;
  Bottom S <= x by YELLOW_0:44;
  then Bottom S in (the carrier of R) /\ (the carrier of S) by A1,A2,
WAYBEL_0:def 19;
  hence thesis by XBOOLE_0:def 4;
end;
