reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem
  for T1,T2, W1,W2 being Tree st tree(T1,T2) = tree(W1,W2) holds
  T1 = W1 & T2 = W2
proof
  let T1,T2, W1,W2 be Tree such that
A1: tree(T1,T2) = tree(W1,W2);
  now
    let p;
    p in T1 iff <*0*>^p in tree(T1,T2) by Th69;
    hence p in T1 iff p in W1 by A1,Th69;
  end;
  hence for p being FinSequence of NAT holds p in T1 iff p in W1;
  let p be FinSequence of NAT;
  p in T2 iff <*1*>^p in tree(T1,T2) by Th70;
  hence thesis by A1,Th70;
end;
