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
  w^t is_a_proper_prefix_of w^s implies t is_a_proper_prefix_of s
proof
  assume
A1: w^t is_a_proper_prefix_of w^s;
then  w^t is_a_prefix_of w^s;
  then consider a be FinSequence such that
A2: w^s = w^t^a by Th1;
 w^t^a = w^(t^a) by FINSEQ_1:32;
then  s= t^a by A2,FINSEQ_1:33;
then  t is_a_prefix_of s by Th1;
  hence thesis by A1;
end;
