
theorem Th16:
  for L being up-complete non empty Poset
  for D being non empty finite directed Subset of L holds sup D in D
proof
  let L be up-complete non empty Poset;
  let D be non empty finite directed Subset of L;
  D c= D;
  then consider d being Element of L such that
A1: d in D and
A2: d is_>=_than D by WAYBEL_0:1;
A3: ex_sup_of D,L by WAYBEL_0:75;
  then
A4: sup D is_>=_than D by YELLOW_0:30;
A5: sup D <= d by A2,A3,YELLOW_0:30;
  sup D >= d by A1,A4;
  hence thesis by A1,A5,ORDERS_2:2;
end;
