(9.109) (Other schools) “When an analytical mind realizes an object to be non-truly existent,
Another analytical mind must analyze that mind to realize that it too is non-truly existent.
That analytical mind, in turn, must be analyzed by another,
And so the process is endless, which is absurd.”

The Prasangikas say that we need to realize the emptiness of our mind. All of our contaminated karma is stored on our very subtle mind and by realizing the emptiness of our very subtle mind, we purify our mind of all of the previously accumulated contaminated karma stored on that mind. Here, the other schools say it is impossible to realize the emptiness of both the object and the mind because there needs to be an additional mind that realizes the emptiness of the mind that realizes the emptiness of the object. And there would then need to be another mind that realizes the emptiness of that mind. For everything to be empty and for that emptiness to be knowable there would have to be an infinite regression of minds knowing the emptiness of the other minds, which is impossible. Thus, they say the Prasangika view is absurd.

(9.110) When a valid mind directly realizes the lack of true existence of all phenomena,
The true existence of that mind is implicitly negated at the same time.
This non-true existence of both subject and object
Is also called the “natural state of nirvana”.

One of the fundamental Prasangika tenets is the mutual dependence between subject and object.  They are distinct things, but they exist in mutual dependence upon one another.  In other words, you can’t have an object without a mind and you cannot have a mind without an object.  If you could, then either the mind or its object would exist inherently. 

The Prasangikas escape the problem of an infinite regression of minds by pointing out the mutual dependence of the subject mind and its object is the valid reason establishing the emptiness of both subject and object simultaneously. Thus, we do not need to realize the emptiness of the mind realizing the emptiness of the mind realizing the emptiness of the object because we can establish the emptiness of the mind directly through realizing the mutual dependence between subject and object. This one valid reason establishes the emptiness of both and therefore the emptiness of both can be realized validly. Again, the logic here is simple. If an object truly existed, it would exist independently of all other phenomena. But if the mind and its object arise in mutual dependence upon one another, which Shantideva’s arguments established above, then they clearly cannot exist independently because they have a dependent relationship.

Shantideva is also saying that the subject/object distinction is likewise merely imputed.  The two are actually the same entity, but nominally distinct, like two sides of the same coin.  This has important practical implications because it means if you make the mind pure, then all the objects of mind will necessarily be pure.  And if you engage only pure objects, all minds you generate will necessarily be pure because they must have the same nature.

(9.111abc) Despite your attempts, you Chittamatrins are unable to establish
The true existence of the apprehending mind and the object apprehended.
(Chittamatrin) “On the contrary, forms, for example, are truly existent because consciousness apprehends them to be so.”

Both the Chittamatrins and the Prasangikas agree for an object to exist, it must be able to be established by a valid mind. The Prasangikas say that the Chittamatrins are unable to establish the true existence of the mind with a valid reason. Therefore, the Prasangikas conclude that the truly existent mind asserted by the Chittamatrins does not exist at all.

The Chittamatrins reply by saying the truly existent mind can by established by saying the objects known to that mind are truly existent, therefore the mind that knows them is also truly existent. The true existence of its object establishes the true existence of the mind knowing it.