 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 Thflat01:
  for P being non empty flat Poset,
      K being non empty Chain of P holds
    ex a being Element of P st K = {a} or K = {Bottom P, a}
  proof
    let P be non empty flat Poset,
        K be non empty Chain of P;
    set z = Bottom P;
    per cases;
      suppose ex a being Element of K st a <> z;
        then consider a being Element of K such that B1:a <> z;
        take a;
D1:     x in K implies x = z or x = a
        proof
          assume
E1:       x in K;
          then reconsider x as Element of P;
          x<=a or a<=x by E1,RELAT_2:def 7,ORDERS_2:def 7;
          hence thesis by Lemflat01,B1;
        end;
        K = {a} or K = {z, a}
          proof
          per cases;
            suppose
C1:           z in K;
              x = z or x = a implies x in K by C1;
              hence thesis by D1,TARSKI:def 2;
            end;
            suppose
C2:           not z in K;
D1:           x in K implies x = a
              proof
                assume
E1:             x in K;
                reconsider x as Element of P by E1;
                x<=a or a<=x by E1,RELAT_2:def 7,ORDERS_2:def 7;
                hence thesis by B1,E1,C2,Lemflat01;
              end;
              for x st x = a holds x in K;
              hence thesis by D1,TARSKI:def 1;
            end;
          end;
          hence thesis;
        end;
      suppose
A1:     not ex a being Element of K st a <> z;
        take z;
B1:     x in K implies x = z by A1;
        x = z implies x in K
        proof
          assume
C1:       x = z;
          the Element of K = z by A1;
          hence thesis by C1;
        end;
        hence thesis by B1,TARSKI:def 1;
      end;
  end;
