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 Th1:
  n <> m implies not <*n*>,<*m*>^s are_c=-comparable
proof
  assume
A1: n<>m;
  assume
A2: <*n*>,<*m*>^s are_c=-comparable;
  per cases by A2;
  suppose
    <*n*> is_a_prefix_of <*m*>^s;
    then
A3: ex a be FinSequence st <*m*>^s = <*n*>^a by TREES_1:1;
    m = (<*m*>^s).1 by FINSEQ_1:41
      .= n by A3,FINSEQ_1:41;
    hence contradiction by A1;
  end;
  suppose
    <*m*>^s is_a_prefix_of <*n*>;
    then consider a be FinSequence such that
A4: <*n*> = <*m*>^s^a by TREES_1:1;
    n = (<*m*>^s^a).1 by A4
      .= (<*m*>^(s^a)).1 by FINSEQ_1:32
      .= m by FINSEQ_1:41;
    hence contradiction by A1;
  end;
end;
