
theorem
  for L being Semilattice, x, y being Element of InclPoset Ids L for x1,
  y1 being Subset of L st x = x1 & y = y1 holds x "/\" y = x1 "/\" y1
proof
  let L be Semilattice, x, y be Element of InclPoset Ids L, x1, y1 be Subset
  of L;
  assume
A1: x = x1 & y = y1;
  then
A2: x1 is lower & y1 is lower by YELLOW_2:41;
  thus x "/\" y = x /\ y by YELLOW_2:43
    .= x1 "/\" y1 by A1,A2,Th50;
end;
