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;
reserve fT,fT1 for finite Tree;
reserve t for Element of T;
reserve w for FinSequence;
reserve t1,t2 for Element of T;
reserve s,t for FinSequence;

theorem
  n <> m implies not <*n*> is_a_prefix_of <*m*>^s
proof
  assume
A1: n <> m;
  assume <*n*> is_a_prefix_of <*m*>^s;
then A2: ex a be FinSequence st <*m*>^s = <*n*>^a by Th1;
 m = (<*m*>^s).1 by FINSEQ_1:41
    .= n by A2,FINSEQ_1:41;
  hence contradiction by A1;
end;
