
theorem Th8:
  for T be Tree holds T = {0,1}* iff for t be Element of T holds
  succ t = { t^<* 0 *>, t^<* 1 *> }
proof
  let T be Tree;
  thus T = {0,1}* implies for t be Element of T holds succ t = { t^<* 0 *>, t^
  <* 1 *> }
  proof
    assume
A1: T = {0,1}*;
    let t be Element of T;
    T is binary & not t in Leaves T by A1,Th3,Th4;
    hence thesis by BINTREE1:def 2;
  end;
  assume
A2: for t be Element of T holds succ t = { t^<* 0 *>, t^<* 1 *> };
  thus T = {0,1}*
  proof
    thus T c= {0,1}*
    proof
      let x be object;
      assume
A3:   x in T;
      T is binary by A2,Th7;
      then x is FinSequence of {0,1} by A3,Th2;
      hence thesis by FINSEQ_1:def 11;
    end;
    defpred P[FinSequence] means $1 in T;
    let x be object;
    assume x in {0,1}*;
    then
A4: x is FinSequence of {0,1} by FINSEQ_1:def 11;
A5: for p be FinSequence of {0,1} for n be Element of {0,1} st P[p] holds
    P[p^<*n*>]
    proof
      let p be FinSequence of {0,1};
      let n be Element of {0,1};
A6:   n = 0 or n = 1 by TARSKI:def 2;
      assume p in T;
      then reconsider p1 = p as Element of T;
      succ p1 = { p1^<* 0 *>, p1^<* 1 *> } by A2;
      then p^<*n*> in succ p1 by A6,TARSKI:def 2;
      hence thesis;
    end;
A7: P[<*> {0,1}] by TREES_1:22;
    for p be FinSequence of {0,1} holds P[p] from FINSEQ_2:sch 2(A7, A5);
    hence thesis by A4;
  end;
end;
