Help On Transitivity Proof

Suppose that [X] <=TP [Y].  This implies the existence of an
algorithm Al[X|Y] for solving instances of X and three
polynomials p, q, and r with the following properties:

1. Al[X|Y] calls an algorithm for solving instances of Y as a
subroutine no more than p(N) times, where N is the size of the
instance of X that Al[X|Y] is working on.

2.  Whenever Al[X|Y] calls the algorithm for Y, the size of the
instance of problem Y that Al[X|Y] gives as the input is no more
than q(N).

3.  Al[X|Y] would take no more than r(N) units of time, if each
call that Al[X|Y] made to the algorithm for Y could be done at
unit cost.

If also [Y] <=TP [Z], then there is an algorithm Al[Y|Z] for
solving instances of Y and three polynomials p', q', and r'
with the following properties:

4. Al[Y|Z] calls an algorithm for solving instances of Z as a
subroutine no more than p'(M) times, where M is the size of the
instance of Y that Al[Y|Z] is working on.

5.  Whenever Al[Y|Z] calls the algorithm for Z, the size of the
instance of problem Z that Al[Y|Z] gives as the input is no more
than q'(M).

6.  Al[Y|Z] would take no more than r'(M) units of time, if each
call that Al[Y|Z] made to the algorithm for Z could be done at
unit cost.

To show that the relation <=TP is transitive, we need to use what 
is given above to prove the following:

7.  There is an algorithm Al[X|Z] for solving instances of X and
three polynomials p", q", and r" with the following properties:

8. Al[X|Z] calls an algorithm for solving instances of Z as a
subroutine no more than p"(N) times, where N is the size of the
instance of X that Al[X|Z] is working on.

9.  Whenever Al[X|Z] calls the algorithm for Z, the size of the
instance of problem Z that Al[X|Z] gives as the input is no more
than q"(N).

10.  Al[X|Z] would take no more than r"(N) units of time, if each
call that Al[X|Z] made to the algorithm for Z could be done at
unit cost.

If you like, you can simplify the "math" by assuming that all of 
the polynomials mentioned above are of the simple form KN^s -- 
a constant K times the s-th power of N.