Sentence examples for proof of cases from inspiring English sources

Case IV: We suppose (omega_{q}=|x_{q}|), (omega_{p}=|y_{p}|), similar to the proof of Cases I, II, III, we can get biglvert {sigma^{2} - vert {a_{pp} } vert ^{2} }bigrvert biglvert {sigma ^{2} - vert {a_{qq} } vert ^{2} }bigrvert le bigl vert {a_{pp} } vert c_{p} (A) + sigma r_{p} (A) bigr) bigl( vert {a_{qq} } vert c_{q} (A) + sigma r_{q} (A) bigr).

Proof of Case I.

Proof of Case  III.

Proof of Case II.

The proof of case is similar.

It is similar to the proof of case 1.

Therefore, we conclude the proof of case (i).

end{aligned} (21) This completes the proof of case (i).

(24) The proof of Case I is complete.

Returning to the proof of Case C, from (1.6), (2.24).

This completes the proof of case (2) in Theorem 1.3.