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 L being Level of W holds L is AntiChain_of_Prefixes of W
proof
  let L be Level of W;
  consider n being Nat such that
A1: L = { w: len w = n } by Def4;
 L is AntiChain_of_Prefixes-like
  proof
    thus for x st x in L holds x is FinSequence
    proof
      let x;
      assume x in L;
then    x is Element of W;
      hence thesis;
    end;
    let v1,v2;
    assume v1 in L;
then A2: ex w1 be Element of W st v1 = w1 & len w1 = n by A1;
    assume v2 in L;
then  ex w2 be Element of W st v2 = w2 & len w2 = n by A1;
    hence thesis by A2,TREES_1:4;
  end;
  hence thesis by TREES_1:def 11;
end;
