Hola Gabor

The sentence you are showing:

"Entonces se puede demostrar que la afirmación es verdadera."

is not an impersonal sentence, it is a passive using "se" .

In the lesson we are explaining how this structure "se puede + infinitive" is a type of impersonal sentence, but for that, it needs verbs that are intransitive (no direct objects).

Here you have a direct object in the form of a subordinate clause:

"que la afirmación es verdadera".

Imagine that we substitute that clause with "algo", so we'd say:

"Entonces se puede demostrar algo"

You can turn this into a passive sentence saying:

Entonces algo puede ser demostrado. 

This is what you got with that translation: 

"Then it can be demonstrated that the statement is true."

However, sometimes, and to make it sound more natural, you can see a translation using an impersonal form: "Then one can demonstrate..." (this is not passive)

In the lesson you saw, if you go to the bottom, you find a link to another lesson on this same structure but using "se puede/n + infinitive". That lesson is about this same struccture but being passive sentences. Here you can use the plural when you have a plural object. For example:

Entonces se pueden demostrar dos cosas.

This is another passive "se" sentence. The verb needs agreement with the object (dos cosas)

I hope this clarifes it a bit. Just so you know, this is a topic (impersonal vs passive se) where not all grammarians agree, so it's always a "noisy" topic. 

Saludos

Inma