 reserve a,Z1,Z2,Z3 for set,
         x,y,z for object,
         k for Nat;
 reserve S for RelStr;
 reserve P,Q for non empty flat Poset;
 reserve p,p1,p2 for Element of P;
 reserve K for non empty Chain of P;

theorem Thflat0502:
  K = {Bottom P, p} implies sup K = p
  proof
    set z = Bottom P;
    assume
A0: K = {z,p};
A1: ex_sup_of K,P by Lemflat02;
    z <= p & p <= p by YELLOW_0:44; then
A2: K is_<=_than p by YELLOW_0:8,A0;
    for p1 st K is_<=_than p1 holds p <= p1 by A0,YELLOW_0:8;
    hence thesis by YELLOW_0:def 9,A1,A2;
  end;
