reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;

theorem
  for A being AntiChain_of_Prefixes of W, C being Chain of W
  ex w st A /\ C c= {w}
proof
  let A be AntiChain_of_Prefixes of W, C be Chain of W;
A1: now
    let p,q;
    assume
A2: p in A /\ C & q in A /\ C;
then A3: p in A & q in A by XBOOLE_0:def 4;
 p in C & q in C by A2,XBOOLE_0:def 4;
then  p,q are_c=-comparable by Def3;
    hence p = q by A3,TREES_1:def 10;
  end;
  set w = the Element of W;
 now per cases;
    suppose
   A /\ C = {};
then    A /\ C c= {w};
      hence thesis;
    end;
    suppose
A4:   A /\ C <> {};
      set x = the Element of A /\ C;
  x in C by A4,XBOOLE_0:def 4;
      then reconsider x as Element of W;
      take x;
      thus A /\ C c= {x}
      proof
        let y be object;
        assume
A5:    y in A /\ C;
then     y is Element of W;
        then reconsider y9 = y as FinSequence of NAT;
    x = y9 by A1,A5;
        hence thesis by TARSKI:def 1;
      end;
    end;
  end;
  hence thesis;
end;
