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 Th60:
  for T being Tree, x being object holds x in ^T iff x = {} or
  ex p st p in T & x = <*0*>^p
proof
  let T be Tree;
  let x;
  set p = <*T*>;
A1: len p = 1 by FINSEQ_1:40;
A2: p.1 = T;
  thus x in ^T & x <> {} implies ex p st p in T & x = <*0*>^p
  proof
    assume that
A3: x in ^T and
A4: x <> {};
    consider n, q such that
A5: n < len p and
A6: q in p.(n+1) and
A7: x = <*n*>^q by A3,A4,Def15;
    0+1 = 1;
    then n = 0 by A1,A5,NAT_1:13;
    hence thesis by A6,A7;
  end;
  0 < 0+1;
  hence thesis by A1,A2,Def15;
end;
