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;

theorem Th11:
  for p,q being FinSequence holds
  p in ProperPrefixes q iff p is_a_proper_prefix_of q
proof
  let p,q be FinSequence;
  thus p in ProperPrefixes q implies p is_a_proper_prefix_of q
  proof
    assume p in ProperPrefixes q;
then
 ex r being FinSequence st p = r & r is_a_proper_prefix_of q by Def2;
    hence thesis;
  end;
  thus thesis by Def2;
end;
