
theorem Th3:
  for T be Tree st T = {0,1}* holds T is binary
proof
  let T be Tree;
  assume
A1: T = {0,1}*;
  now
    let t be Element of T;
    assume not t in Leaves T;
    { t^<*n*> where n is Nat : t^<*n*> in T } = { t^<* 0 *>, t^
    <* 1 *> }
    proof
      thus {t^<*n*> where n is Nat : t^<*n*> in T} c= { t^<* 0 *>,
      t^<* 1 *> }
      proof
        let x be object;
        assume x in { t^<*n*> where n is Nat : t^<*n*> in T };
        then consider n be Nat such that
A2:     x = t^<*n*> and
A3:     t^<*n*> in T;
        reconsider tn = t^<*n*> as FinSequence of ({0,1} qua set) by A1,A3,
FINSEQ_1:def 11;
        len t + 1 in Seg (len t + 1) by FINSEQ_1:4;
        then len t + 1 in Seg len tn by FINSEQ_2:16;
        then len t + 1 in dom tn by FINSEQ_1:def 3;
        then tn/.(len t + 1) = (t^<*n*>).(len t + 1) by PARTFUN1:def 6
          .= n by FINSEQ_1:42;
        then n = 0 or n = 1 by TARSKI:def 2;
        hence thesis by A2,TARSKI:def 2;
      end;
      let x be object;
A4:   t is FinSequence of {0,1} by A1,FINSEQ_1:def 11;
      rng <* 1 *> c= {0,1}
      proof
        let z be object;
        assume z in rng <* 1 *>;
        then z in {1} by FINSEQ_1:38;
        then z = 1 by TARSKI:def 1;
        hence thesis by TARSKI:def 2;
      end;
      then <* 1 *> is FinSequence of {0,1} by FINSEQ_1:def 4;
      then t^<* 1 *> is FinSequence of {0,1} by A4,Lm1;
      then
A5:   t^<* 1 *> in T by A1,FINSEQ_1:def 11;
      assume x in { t^<* 0 *>, t^<* 1 *> };
      then
A6:   x = t^<* 0 *> or x = t^<* 1 *> by TARSKI:def 2;
      rng <* 0 *> c= {0,1}
      proof
        let z be object;
        assume z in rng <* 0 *>;
        then z in {0} by FINSEQ_1:38;
        then z = 0 by TARSKI:def 1;
        hence thesis by TARSKI:def 2;
      end;
      then <* 0 *> is FinSequence of {0,1} by FINSEQ_1:def 4;
      then t^<* 0 *> is FinSequence of {0,1} by A4,Lm1;
      then t^<* 0 *> in T by A1,FINSEQ_1:def 11;
      hence thesis by A5,A6;
    end;
    hence succ t = { t^<* 0 *>, t^<* 1 *> } by TREES_2:def 5;
  end;
  hence thesis by BINTREE1:def 2;
end;
