
theorem Th4:
  for T be Tree st T = {0,1}* holds Leaves T = {}
proof
A1: {0} c= BOOLEAN
  proof
    let z be object;
    assume z in {0};
    then z = FALSE by TARSKI:def 1;
    hence thesis;
  end;
  let T be Tree;
  assume
A2: T = {0,1}*;
  assume Leaves T <> {};
  then consider x be object such that
A3: x in Leaves T by XBOOLE_0:def 1;
  reconsider x1 = x as Element of T by A3;
  T is binary by A2,Th3;
  then
A4: x1 is FinSequence of BOOLEAN by Th2;
  then reconsider x1 = x as FinSequence of NAT;
  set y1 = x1 ^ <* 0 *>;
  0 in {0} by TARSKI:def 1;
  then <* 0 *> is FinSequence of BOOLEAN by A1,Lm2;
  then y1 is FinSequence of BOOLEAN by A4,Lm1;
  then
A5: y1 in T by A2,FINSEQ_1:def 11;
  x1 is_a_proper_prefix_of y1 by TREES_1:8;
  hence contradiction by A3,A5,TREES_1:def 5;
end;
