
theorem
  for L being with_suprema Poset for x, y being Element of InclPoset Ids
  L for X, Y being Subset of L st x = X & y = Y holds x "\/" y = downarrow (X
  "\/" Y)
proof
  let L be with_suprema Poset, x, y be Element of InclPoset Ids L, X, Y be
  Subset of L such that
A1: x = X and
A2: y = Y;
  reconsider X1 = X, Y1 = Y as non empty directed Subset of L by A1,A2,
YELLOW_2:41;
  reconsider d = downarrow (X1 "\/" Y1) as Element of InclPoset Ids L by
YELLOW_2:41;
  Y c= d by Th29;
  then
A3: y <= d by A2,YELLOW_1:3;
  X c= d by Th29;
  then x <= d by A1,YELLOW_1:3;
  then d <= d & x "\/" y <= d "\/" d by A3,YELLOW_3:3;
  then x "\/" y <= d by YELLOW_0:24;
  hence x "\/" y c= downarrow (X "\/" Y) by YELLOW_1:3;
  consider Z being Subset of L such that
A4: Z = {z where z is Element of L: z in x or z in y or ex a, b being
  Element of L st a in x & b in y & z = a "\/" b} and
  ex_sup_of {x,y},InclPoset(Ids L) and
A5: x "\/" y = downarrow Z by YELLOW_2:44;
  X "\/" Y c= Z
  proof
    let q be object;
    assume q in X "\/" Y;
    then ex s, t being Element of L st q = s "\/" t & s in X & t in Y;
    hence thesis by A1,A2,A4;
  end;
  hence thesis by A5,YELLOW_3:6;
end;
