reserve A, B, X, Y for set;

theorem
  for L1 being sup-Semilattice, L2 being non empty RelStr for X, Y being
Subset of L1, X1, Y1 being Subset of L2 st the RelStr of L1 = the RelStr of L2
  & X = X1 & Y = Y1 holds X "\/" Y = X1 "\/" Y1
proof
  let L1 be sup-Semilattice, L2 be non empty RelStr, X, Y be Subset of L1, X1,
  Y1 be Subset of L2 such that
A1: the RelStr of L1 = the RelStr of L2 and
A2: X = X1 & Y = Y1;
  set XY = { x "\/" y where x, y is Element of L1 : x in X & y in Y }, XY1 = {
  x "\/" y where x, y is Element of L2 : x in X1 & y in Y1 };
A3: XY = XY1
  proof
    hereby
      let a be object;
      assume a in XY;
      then consider x, y being Element of L1 such that
A4:   a = x "\/" y and
A5:   x in X & y in Y;
      reconsider x1 = x, y1 = y as Element of L2 by A1;
      a = x1 "\/" y1 by A1,A4,Th10;
      hence a in XY1 by A2,A5;
    end;
    let a be object;
    assume a in XY1;
    then consider x, y being Element of L2 such that
A6: a = x "\/" y and
A7: x in X1 & y in Y1;
    reconsider x1 = x, y1 = y as Element of L1 by A1;
    a = x1 "\/" y1 by A1,A6,Th10;
    hence thesis by A2,A7;
  end;
  thus X "\/" Y = XY by YELLOW_4:def 3
    .= X1 "\/" Y1 by A3,YELLOW_4:def 3;
end;
