Sentence examples for lines of proofs from inspiring English sources

With these lines of proofs, Algorithm AWCR generates a point sequence and every point of the point sequence consists of the K non-negative numbers, e.g., S 1 ( n ), …, S K ( n ).

Its proof completely follows the lines of proof of Theorem 1 and hence it is omitted.

The lines of proof are not effected and therefore the proofs are omitted.

Hence similar to the lines of proof of Theorems 5 and 7 our result follows.

Follows the lines of proof of Theorem 3.9 [8], so is omitted.

Following the lines of proof of Theorem 2.1, we get that problem (1.1) has a solution in W 0 1, p.

Moreover, following the lines of proof of the lemma in [4], we get ∫ Ω u 1 r d x < ∞ if and only if r > − 1.  □.

By following the lines of proof as above, and passing to a subsequence if necessary, we know that there is such that as j → ∞.

From (ii), by following the same lines of proof as in Theorem 3.9, we get that (u in operatorname{Fix}(T)).

By following the lines of proof as above (passing to a subsequence if necessary), we know that there exists u ˜ ∈ GVI K ( A, g, S ) such that g ( u α j ) → g ( u ˜ ) as j → ∞.

There is no doubt that the requested experiments could provide additional examples of the concepts we demonstrated, but they would not add new lines of proof.