 reserve a,Z1,Z2,Z3 for set,
         x,y,z for object,
         k for Nat;

theorem Thsup02:
  for P being chain-complete non empty Poset,
      L being non empty Chain of P
  for p1 being Element of P st
     (for p being Element of P st p in L holds p<=p1) holds sup L <= p1
  proof
    let P be chain-complete non empty Poset;
    let L be non empty Chain of P;
    let p1 be Element of P;
    assume for p being Element of P st p in L holds p<=p1; then
A1: L is_<=_than p1;
    ex_sup_of L,P by POSET_1:def 1;
    hence thesis by YELLOW_0:def 9,A1;
  end;
