Sentence examples for proved the same as from inspiring English sources

Suppose that u N j is a solution of problem (31), then there exist positive constants c 12, c 13, c 14, c 15, c 16 depending only on k, a, b, T and ∥ u 0 ∥ H 2 such that ∥ u N j ∥ ≤ c 12, ∥ u N x j ∥ ≤ c 13, ∥ u N x x j ∥ ≤ c 14. Furthermore, we have sup x ∈ [ 0, 1 ] | u N j | ≤ c 15, sup x ∈ [ 0, 1 ] | u N x j | ≤ c 16. Proof It can be proved the same as Lemmas 2.1-2.3 2.1-2.3

The fact that (rho _G(n ge f(n)) is proved the same way as in the proof of Theorem 3.12 (one only needs to replace the product by operation * everywhere in that proof).

In [9] Yuan and Ou proved the same conclusions as in [2] by using the linking theorem over cones by Degiovanni and Lancelotti (see [18]), where the eigenvalues of (-Delta_{p}) are defined by the (Z_{2} -cohomological index.

Therefore, W (z ) must have a zero at z = 0. □ The sufficiency can be proved the same way as the sufficiency part of Theorem 4 by replacing z 0 there with 0. Necessity.

Under  Assumption 1  and assuming that (A, B, C, D ) is minimal, then V (Z ) has a zero at Z = 0 if and only if W (z ) has a zero at z = 0. Proof The sufficiency can be proved the same way as the sufficiency part of Theorem 4 by replacing z 0 there with 0. Necessity.

The designated fragments in Fig.  2A were cloned and sequenced, proving the same sequence as the target genes published online (accession no. NC_000853).

However, we manage to prove the same good characteristics as in the LO case.

In Leigh (2012) it is shown that the theory proves the same Π02 arithmetical statements as classical FS.

Like the typed theory in Section 3.2 this theory is does not prove certain generalizations but proves the same T-free sentences as the strong type-free compositional Kripke-Feferman theory below (Halbach 2009).

It is possible to prove the same type of results as for the SIR epidemic.

Since the method of proving is the same as that of Theorem 2, it is omitted.