
theorem Th14:
  for L being upper-bounded Semilattice, X being Subset of L holds
  {Top L} "/\" X = X
proof
  let L be upper-bounded Semilattice, X be Subset of L;
A1: {Top L} "/\" X = {Top L "/\" y where y is Element of L: y in X} by
YELLOW_4:42;
  thus {Top L} "/\" X c= X
  proof
    let q be object;
    assume q in {Top L} "/\" X;
    then ex y being Element of L st q = Top L "/\" y & y in X by A1;
    hence thesis by WAYBEL_1:4;
  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 = Top L "/\" y by WAYBEL_1:4;
  hence thesis by A1;
end;
