
theorem Th4:
  for L being up-complete sup-Semilattice for A, B being non empty
  directed Subset of L holds sup (A "\/" B) = sup A "\/" sup B
proof
  let L be up-complete sup-Semilattice, A, B be non empty directed Subset of L;
A1: ex_sup_of B,L by WAYBEL_0:75;
  then
A2: B is_<=_than sup B by YELLOW_0:30;
A3: ex_sup_of A,L by WAYBEL_0:75;
  then A is_<=_than sup A by YELLOW_0:30;
  then ex_sup_of A "\/" B,L & A "\/" B is_<=_than sup A "\/" sup B by A2,
WAYBEL_0:75,YELLOW_4:27;
  then
A4: sup (A "\/" B) <= sup A "\/" sup B by YELLOW_0:def 9;
  B is_<=_than sup (A "\/" B) by Th3;
  then
A5: sup B <= sup (A "\/" B) by A1,YELLOW_0:30;
  A is_<=_than sup (A "\/" B) by Th3;
  then sup A <= sup (A "\/" B) by A3,YELLOW_0:30;
  then
A6: sup A "\/" sup B <= sup (A "\/" B) "\/" sup (A "\/" B) by A5,YELLOW_3:3;
  sup (A "\/" B) <= sup (A "\/" B);
  then sup (A "\/" B) "\/" sup (A "\/" B) = sup (A "\/" B) by YELLOW_0:24;
  hence thesis by A4,A6,ORDERS_2:2;
end;
