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,r being FinSequence st q^r in ProperPrefixes p holds
  q in ProperPrefixes p
proof
  let p,q,r be FinSequence;
  assume
A1: q^r in ProperPrefixes p;
A2: q is_a_prefix_of q^r by Th1;
 q^r is_a_proper_prefix_of p by A1,Th11;
 then q is_a_proper_prefix_of p by A2,XBOOLE_1:59;
  hence thesis by Th11;
end;
