 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 Thflat05:
  for f being Function of P,Q holds
    ex a being Element of P st
      (K = {a} & f.:K = {f.a}) or
      (K = {Bottom P, a} & f.:K = {f.(Bottom P), f.a})
  proof
    let f be Function of P,Q;
    consider a being Element of P such that
A1: K = {a} or K = {Bottom P, a} by Thflat01;
    take a;
    set z = Bottom P;
A3: the carrier of P = dom f by FUNCT_2:def 1;
    K = {a,a} or K = {Bottom P, a} by A1,ENUMSET1:29;
    then (K = {a,a} & f.:K = {f.a,f.a}) or
         (K = {Bottom P, a} & f.:K = {f.(Bottom P), f.a}) by A3,FUNCT_1:60;
    hence thesis by ENUMSET1:29;
  end;
