Symplectic approach to canonical transformations

Hi, I'm studying classical mechanics via Goldstein's book. And at the beginning of the chapter on the symplectic approach to canonical transformations he writes (long citation ahead):

Third Edition, Page 381
CITATION BEGIN

By way of intruduction to this approach, let us consider a restricted canonical transformation, that is, one in which time does not appear in the equations of transformations:

(9.44)
[itex]Q_{i}=Q_{i}(q,p)[/itex]
[itex]P_{i}=P_{i}(q,p)[/itex]

We know that the Hamiltonian function does not change in such a transformation. The time derivative of [itex]Q_{i}(q,p)[/itex] on the basis of Eqs. 9.44, is to be found as

(9.45)
[itex]\dot{Q_{i}}= \frac{\partial Q_{i}}{\partial q_{j}} \dot{q_{j}} + \frac{\partial Q_{i}}{\partial p_{j}} \dot{p_{j}} = \frac{\partial Q_{i}}{\partial q_{j}} \frac{\partial H}{\partial p_{j}} - \frac{\partial Q_{i}}{\partial p_{j}} \frac{\partial H}{\partial q_{j}} [/itex]

On the other hand, the inverses of Eqs. 9.44

(9.46)
[itex]q_{j}= q_{j}(Q,P)[/itex],
[itex]p_{j}= p_{j}(Q,P)[/itex]

enables us to consider [itex]H(q,p,t)[/itex] as a function of [itex]Q[/itex] and [itex]P[/itex] and to form the partial derivative

(9.47)

[itex] \frac{\partial H(p_{j}(Q_{i},P_{i}), q_{j}(Q_{i},P_{i}))}{\partial P_{i}} = \frac{\partial H}{\partial p_{j}} \frac{\partial p_{j}}{\partial P_{i}} + \frac{\partial H}{\partial q_{j}} \frac{\partial q_{j}}{\partial P_{i}}[/itex]

Comparing Eqs. (9.45) and (9.47), it can be concluded that

[itex]\dot{Q_{i}} = \frac{\partial H}{\partial P_{i}}[/itex]

that is, the transformation is canonical, only if

(9.48a)
[itex](\frac{\partial Q_{i}}{\partial q_{j}})_{q,p} = (\frac{\partial p_{j}}{\partial P_{i}})_{Q,P} , (\frac{\partial Q_{i}}{\partial p_{j}})_{q,p} = (\frac{\partial q_{j}}{\partial P_{i}})_{Q,P}[/itex]

CITATION END

I don't understand what he did here:

"
Comparing Eqs. (9.45) and (9.47), it can be concluded that

[itex]\dot{Q_{i}} = \frac{\partial H}{\partial P_{i}}[/itex]

that is, the transformation is canonical, only if

(9.48a)
[itex](\frac{\partial Q_{i}}{\partial q_{j}})_{q,p} = (\frac{\partial p_{j}}{\partial P_{i}})_{Q,P} , (\frac{\partial Q_{i}}{\partial p_{j}})_{q,p} = (\frac{\partial q_{j}}{\partial P_{i}})_{Q,P}[/itex]"


How did he derive (9.48a) ?? I'm lost at "Comparing Eqs. 9.45 and 9.47...." What does mean with comparing?? :-/

Greets