
theorem Th42:
  for L be lower-bounded antisymmetric non empty RelStr for S be
  with_bottom Subset of L holds subrelstr S is lower-bounded
proof
  let L be lower-bounded antisymmetric non empty RelStr;
  let S be with_bottom Subset of L;
  Bottom L in S by Def8;
  then reconsider dL = Bottom L as Element of subrelstr S by YELLOW_0:def 15;
  take dL;
  now
    let x be Element of subrelstr S;
    assume x in the carrier of subrelstr S;
    reconsider x1 = x as Element of L by YELLOW_0:58;
    Bottom L <= x1 by YELLOW_0:44;
    hence dL <= x by YELLOW_0:60;
  end;
  hence thesis;
end;
