Adverbs with prepositions - English Language & Usage

Much to my surprise, I've read recently that some adverbs do not inherit prepositional constructions from the adjectives they come from, for example:

"The proof of Theorem 3 is similar to that of Theorem 2"

is OK, but

"Theorem 3 can be proved similarly to Theorem 2"

is not quite correct.

Is that indeed true?

This may be the reason why sometimes one can find "similar" in phrases playing the role of "sentence adverbs", like

Similar to the situation in Section 2, we will now consider etc.

This default of English adverbs causes problems when translating e.g. from Slavonic languages, where adverbs like "similarly", "analogously" connect with prepositions.