reserve
  X,x,y,z for set,
  k,n,m for Nat ,
  f for Function,
  p,q,r for FinSequence of NAT;
reserve p1,p2,p3 for FinSequence;
reserve T,T1 for Tree;

theorem Th19:
  for p,q being FinSequence st p in T & q is_a_prefix_of p holds q in T
proof
  let p,q be FinSequence;
  assume that
A1: p in T and
A2: q is_a_prefix_of p;
  reconsider r = p as Element of T by A1;
A3: ProperPrefixes r c= T by Def3;
 q is_a_proper_prefix_of p or q = p by A2;
then  q in ProperPrefixes p or p = q by Th11;
  hence thesis by A3;
end;
