
theorem
  for L being lower-bounded with_infima antisymmetric RelStr for X being
  non empty Subset of L holds X "/\" {Bottom L} = {Bottom L}
proof
  let L be lower-bounded with_infima antisymmetric RelStr, X be non empty
  Subset of L;
  thus X "/\" {Bottom L} c= {Bottom L} by Th10;
  let q be object;
  assume q in {Bottom L};
  then
A1: X "/\" {Bottom L} = {Bottom L "/\" y where y is Element of L: y in X} &
  q = Bottom L by TARSKI:def 1,YELLOW_4:42;
  consider y being object such that
A2: y in X by XBOOLE_0:def 1;
  reconsider y as Element of X by A2;
  Bottom L "/\" y = Bottom L by WAYBEL_1:3;
  hence thesis by A1;
end;
