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

theorem Thsup01:
  for P being chain-complete non empty Poset,
      L being non empty Chain of P
  for p being Element of P st p in L holds p <= sup L
  proof
    let P be chain-complete non empty Poset;
    let L be non empty Chain of P;
    let p be Element of P;
    assume
A1: p in L;
A2: ex_sup_of L,P by POSET_1:def 1;
    reconsider x = sup L as Element of P;
    L is_<=_than x by YELLOW_0:def 9,A2;
    hence thesis by A1;
  end;
