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;
