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

1.	Under the entry прежде appears the phrase прежде чем as a conjunction, meaning before, which is usually followed by an infinitive. ↑
2.	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–λI матрицы is literally "of the adjoint to A–λI matrix" and is somewhat reminiscent of a construction in German. Note that the – between A and λI 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. ↑

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Для каждой строго возрастающей¹ последовательности σ индексов множество A_σ открыто² в L_n+1,p+1 и насыщено² по Δ_n,p, поэтому его³ канонический образ C_σ

в P_n,p, является открытым множеством,⁴ гомеоморфным фактормножеству множества A_σ по отношению эквивалентности⁵ θ_σ, индуцируемому⁶ в A_σ отношением Δ_n,p.

For every strictly increasing sequence of indices σ, the set A_σ is open in L_n

+1,p+1 and saturated for Δ_n,p; hence its canonical image C_σ in P_n,p is an open set (which is)⁷ homeomorphic to the quotient of the set A_σ by the equivalence relation θ_σ induced in A_σ by the relation Δ_n,p.

Notes