theorem Th25:
  p is_a_prefix_of q & q in B implies p in B
proof
  assume p is_a_prefix_of q;
then  p is_a_proper_prefix_of q or p = q;
then A1: p in ProperPrefixes q or p = q by TREES_1:def 2;
  assume
A2: q in B;
then  ProperPrefixes q c= B by Def7;
  hence thesis by A1,A2;
end;
