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