
theorem
  for L being lower-bounded with_suprema Poset, X being Subset of L
  holds {Bottom L} "\/" X = X
proof
  let L be lower-bounded with_suprema Poset, X be Subset of L;
A1: {Bottom L} "\/" X = {Bottom L "\/" y where y is Element of L: y in X} by
YELLOW_4:15;
  thus {Bottom L} "\/" X c= X
  proof
    let q be object;
    assume q in {Bottom L} "\/" X;
    then ex y being Element of L st q = Bottom L "\/" y & y in X by A1;
    hence thesis by WAYBEL_1:3;
  end;
  let q be object;
  assume
A2: q in X;
  then reconsider X1 = X as non empty Subset of L;
  reconsider y = q as Element of X1 by A2;
  q = Bottom L "\/" y by WAYBEL_1:3;
  hence thesis by A1;
end;
