reserve x,y,x1,x2,z for set,
  n,m,k for Nat,
  t1 for (DecoratedTree of [: NAT,NAT :]),
  w,s,t,u for FinSequence of NAT,
  D for non empty set;

theorem
  w^t is_a_proper_prefix_of w^s implies t is_a_proper_prefix_of s by TREES_1:49
