Sentence examples for proved the same way from inspiring English sources
We only prove the (operatorname{BPMA}(1)) case here, the (operatorname{BPAR}(1)) case could be proved the same way.
The alternative case, in which we have conclusion (ii) in Lemma 4.4, is proved the same way, using in place of.
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).
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.
Since (ii) can be proved in the same way, we just prove (i).
Proof The first three parts can be proved by the same way as we have proved for contact CR-warped products in [12].
The lower bound is proved in the same way.
The following statement is proved in the same way.
Other cases can be proved in the same way.
The uniqueness of the fixed point can be easily proved in the same way as above.
