
theorem Th3:
  for L being up-complete sup-Semilattice for A, B being non empty
  directed Subset of L holds A is_<=_than sup (A "\/" B)
proof
  let L be up-complete sup-Semilattice, A, B be non empty directed Subset of L;
A1: A "\/" B = { x "\/" y where x, y is Element of L : x in A & y in B } by
YELLOW_4:def 3;
  set b = the Element of B;
  let x be Element of L;
  assume x in A;
  then
A2: x "\/" b in A "\/" B by A1;
  ex xx being Element of L st x <= xx & b <= xx & for c being Element of L
  st x <= c & b <= c holds xx <= c by LATTICE3:def 10;
  then
A3: x <= x "\/" b by LATTICE3:def 13;
  ex_sup_of A "\/" B,L by WAYBEL_0:75;
  then A "\/" B is_<=_than sup (A "\/" B) by YELLOW_0:def 9;
  then x "\/" b <= sup (A "\/" B) by A2;
  hence x <= sup (A "\/" B) by A3,YELLOW_0:def 2;
end;
