
theorem Th2:
  for T be binary Tree for t be Element of T holds t is FinSequence of BOOLEAN
proof
  let T be binary Tree;
  let t be Element of T;
  defpred P[FinSequence] means $1 is Element of T implies rng $1 c= BOOLEAN;
A1: for p be FinSequence of NAT
    for x be Element of NAT st P[p] holds P[p^<*x*>]
  proof
    let p be FinSequence of NAT;
    let x be Element of NAT;
    assume
A2: P[p];
    assume
A3: p^<*x*> is Element of T;
    then reconsider p1 = p as Element of T by TREES_1:21;
    p^<*x*> in { p^<*n*> where n is Nat : p^<*n*> in T } by A3;
    then
A4: p^<*x*> in succ p1 by TREES_2:def 5;
    then not p in Leaves T by BINTREE1:3;
    then succ p1 = { p^<* 0 *>, p^<*1*> } by BINTREE1:def 2;
    then p^<*x*> = p^<* 0 *> or p^<*x*> = p^<*1*> by A4,TARSKI:def 2;
    then x = 0 or x = 1 by FINSEQ_2:17;
    then
A5: x in {0,1} by TARSKI:def 2;
A6: {x} c= BOOLEAN
    by A5,TARSKI:def 1;
    rng <*x*> = {x} by FINSEQ_1:38;
    then (rng p) \/ (rng <*x*>) c= BOOLEAN by A2,A3,A6,TREES_1:21,XBOOLE_1:8;
    hence thesis by FINSEQ_1:31;
  end;
A7: P[<*>NAT] by XBOOLE_1:2;
  for p be FinSequence of NAT holds P[p] from FINSEQ_2:sch 2(A7,A1);
  then rng t c= BOOLEAN;
  hence thesis by FINSEQ_1:def 4;
end;
