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
  for p,q being FinSequence st p is_a_prefix_of q holds
  ProperPrefixes p c= ProperPrefixes q
proof
  let p,q be FinSequence such that
A1: p is_a_prefix_of q;
  let x be object;
  assume
A2: x in ProperPrefixes p;
  then reconsider r = x as FinSequence by Th10;
 r is_a_proper_prefix_of p by A2,Th11;
then  r is_a_proper_prefix_of q by A1,XBOOLE_1:58;
  hence thesis by Th11;
end;
