
theorem Th5:
  for L be with_suprema Poset for X,Y be Subset of L for X9,Y9 be
  Subset of L opp st X = X9 & Y = Y9 holds X "\/" Y = X9 "/\" Y9
proof
  let L be with_suprema Poset;
  let X,Y be Subset of L;
  let X9,Y9 be Subset of L opp;
  assume
A1: X = X9 & Y = Y9;
  thus X "\/" Y c= X9 "/\" Y9
  proof
    let a be object;
    assume a in X "\/" Y;
    then a in { p "\/" q where p,q is Element of L : p in X & q in Y } by
YELLOW_4:def 3;
    then consider x,y be Element of L such that
A2: a = x "\/" y and
A3: x in X & y in Y;
A4: a = x~ "/\" y~ by A2,YELLOW_7:23;
    x~ in X9 & y~ in Y9 by A1,A3,LATTICE3:def 6;
    then
    a in { p "/\" q where p,q is Element of L opp : p in X9 & q in Y9 } by A4;
    hence thesis by YELLOW_4:def 4;
  end;
  let a be object;
  assume a in X9 "/\" Y9;
  then a in { p "/\" q where p,q is Element of L opp : p in X9 & q in Y9 } by
YELLOW_4:def 4;
  then consider x,y be Element of L opp such that
A5: a = x "/\" y and
A6: x in X9 & y in Y9;
A7: a = ~x "\/" ~y by A5,YELLOW_7:24;
  ~x in X & ~y in Y by A1,A6,LATTICE3:def 7;
  then a in { p "\/" q where p,q is Element of L : p in X & q in Y } by A7;
  hence thesis by YELLOW_4:def 3;
end;
