theorem Th43:
  for T being Tree, t being Element of T holds ProperPrefixes t is
  finite Chain of T
proof
  let T be Tree, t be Element of T;
  ProperPrefixes t c= T & for p,q being FinSequence of NAT st p in
ProperPrefixes t & q in ProperPrefixes t holds p,q are_c=-comparable by
TREES_1:18,def 3;
  hence thesis by TREES_2:def 3;
end;
