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 Th8:
  p1 is_a_proper_prefix_of p2^<*x*> implies p1 is_a_prefix_of p2
proof
  assume that
A1: p1 is_a_prefix_of p2^<*x*> and
A2: p1 <> p2^<*x*>;
A3: ex p3 st p2^<*x*> = p1^p3 by A1,Th1;
A4: len p1 <= len(p2^<*x*>) by A1,NAT_1:43;
   len(p2^<*x*>) = len p2 + len <*x*> & len <*x*> = 1 by FINSEQ_1:22,39;
then  len p1 < len p2 + 1 by A1,A2,A4,CARD_2:102,XXREAL_0:1;
then  len p1 <= len p2 by NAT_1:13;
then  ex p3 st p1^p3 = p2 by A3,FINSEQ_1:47;
  hence thesis by Th1;
end;
