A. J. Lohwater's Russian-English Dictionary of the Mathematical Sciences

The infinitive here can be rendered in many ways, the simplest and most usual way being 'Before proving that...'; the less common English expression in the translation has been used to emphasize the use of the imperfective aspect which must be explicit in Russian. ^
3.	Imperfective infinitive is удовлетворять; the conjugation is like выполнять in Appendix D. ^
4.	Perfective form доказать from the imperfective infinitive доказывать (see Appendix D). Although докажем is first-person plural, future indicative, the translation 'we prove' is as correct as 'we shall prove,' since the two renderings are fully equivalent in the context of the sentence. ^
5.	The expression общий наибольший делитель will be found listed in the dictionary after each of the three words in the phrase, since a literal rendering with the same word-order could conceivably lead to confusion, particularly since общий has other meanings. ^
6.	The phrase присоединённой к A– ?I матрицы is literally 'of the adjoint to A–?I matrix' and is somewhat reminiscent of a construction in German. Note that the – between A and ?I cannot be misinterpreted as a substitute for the verb to be, since Soviet mathematical writers are quite careful not to use the dash or hyphen for the verb to be whenever there is any possibility of confusion. ^
7.	The word то is the conjunction then, not the demonstrative pronoun. ^

(2)

Для каждой строго возрастающей¹ последовательности ? индексов множество A_? открыто² в L_{n +1,p+1} и насыщено² по ?_n,p, поэтому его³ канонический образ C_? в P_n,p, является открытым множеством,⁴ гомеоморфным фактормножеству множества A_? по отношению эквивалентности⁵ ?_?, индуцируемому⁶ в A_? отношением ?_n,p.

For every strictly increasing sequence of indices ?, the set A_? is open in L_n+1,p+1 and saturated for ?_n,p; hence its canonical image C_? in P_n,p is an open set (which is)⁷ homeomorphic to the quotient of the set A_? by the equivalence relation ?_? induced in A_? by the relation ?_n,p.

Notes

1.	Present participle, having the form of an adjective; this form is listed as an adjective in the dictionary. ^
2.	Short form, neuter, of открытый (open) indicates a predicative use with the verb 'to be' omitted; note the short form, neuter, of насыщенный (saturated). ^
3.	Note that его is not declined, even though it is linked to образ. ^
4.	The verb являться (to be) is followed by the instrumental case. ^
5.	отношение эквивалентности (lit. relation of equivalence) is the usual way of expressing the English phrase equivalence relation. In Russian, the noun 'equivalence' cannot be used as an adjective without adding an adjectival ending. It is quite possible to form the adjective эквивалентностный (equivalence, pert. to an equivalence) and write эквивалентностное отношение (equivalence relation), but it lacks one essential ingredient: it is not the idiomatic expression for 'equivalence relation' in Russian. ^