
theorem Th60:
  for L being lower-bounded antisymmetric non empty RelStr
  for S being non empty bottom-inheriting full SubRelStr of L
  holds Bottom S = Bottom L
proof
  let L be lower-bounded antisymmetric non empty RelStr;
  let S be non empty bottom-inheriting full SubRelStr of L;
  reconsider s = Bottom L as Element of S by Def19;
  reconsider l = Bottom S as Element of L by YELLOW_0:58;
A1: Bottom L <= l by YELLOW_0:44;
A2: Bottom S <= s by YELLOW_0:44;
  Bottom S >= s by A1,YELLOW_0:60;
  hence thesis by A2,ORDERS_2:2;
end;
