
theorem Th5:
  for t being Tree, p being Element of t holds t|p =
  elementary_tree 0 iff p in Leaves t
proof
  let t be Tree, p be Element of t;
A1: not <*0*> in elementary_tree 0 by TARSKI:def 1,TREES_1:29;
  hereby
    assume t|p = elementary_tree 0;
    then not p^<*0*> in t by A1,TREES_1:def 6;
    hence p in Leaves t by TREES_1:54;
  end;
  assume
A2: p in Leaves t;
  let q be FinSequence of NAT;
  hereby
    assume q in t|p;
    then p^q in t by TREES_1:def 6;
    then
A3: not p is_a_proper_prefix_of p^q by A2,TREES_1:def 5;
    p is_a_prefix_of p^q by TREES_1:1;
    then p^q = p by A3
      .= p^{} by FINSEQ_1:34;
    then q = {} by FINSEQ_1:33;
    hence q in elementary_tree 0 by TREES_1:22;
  end;
  assume q in elementary_tree 0;
  then q = {} by TARSKI:def 1,TREES_1:29;
  hence thesis by TREES_1:22;
end;
