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
  w = {} implies W-level 1 = succ w
proof
  assume
A1: w = {};
  thus W-level 1 c= succ w
  proof
    let x be object;
    assume x in W-level 1;
    then consider w9 such that
A2: x = w9 and
A3: len w9 = 1;
A4: w9 = <*w9.1*> by A3,FINSEQ_1:40;
then  rng w9 = {w9.1} by FINSEQ_1:39;
    then reconsider n = w9.1 as Element of NAT by ZFMISC_1:31;
 w9 = w^<*n*> by A1,A4,FINSEQ_1:34;
    hence thesis by A2;
  end;
  let x be object;
  assume x in succ w;
  then consider i such that
A5: x = w^<*i*> and
A6: w^<*i*> in W;
  reconsider w9 = w^<*i*> as Element of W by A6;
 len <*i*> = 1 & w9 = <*i*> by A1,FINSEQ_1:34,39;
  hence thesis by A5;
end;
