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 Th2:
  n <> m implies not <*n*> is_a_proper_prefix_of <*m*>^s
proof
  assume
A1: n <> m;
  assume <*n*> is_a_proper_prefix_of <*m*>^s;
  then <*n*> is_a_prefix_of <*m*>^s;
  then
A2: ex a be FinSequence st <*m*>^s = <*n*>^a by TREES_1:1;
  m = (<*m*>^s).1 by FINSEQ_1:41
    .= n by A2,FINSEQ_1:41;
  hence contradiction by A1;
end;
