Sentence examples for preceding proofs from inspiring English sources

The proof of this statement is a total analogy with the preceding proofs as consequences of the main statements.

The proof of this statement is a total analogy with the preceding proofs of Corollaries 2 and 3. Thus we omit it.

The proof of this statement is a total analogy with the preceding proofs of Corollaries 2 and 3 (because Corollary 22 is a dual form of Corollary 6).

We recall from the preceding proofs that we can without loss of generality assume (Qle 0) and that we aim to bound the negative eigenvalues ({-lambda _i}) of (mathcal {H}).

Similar to the preceding proof, we obtain that.

The proof of this statement is a total analogy with the preceding proof of Proposition 3.

The preceding proof states that a bias does not affect the stability analysis.

Similar to the preceding proof, we obtain that lim i → ∞ x i = x ¯.

Obviously, the choice of τ, the values of ϱ and ϵ in the preceding proof depends on λ.

For and, as the preceding proof in Theorem 2.1, there exist integers and mappings satisfying (2.5)–(2.7), where are replaced by, and,, respectively, and for some.

Since maps into and restricted to is invertible, we can apply the preceding proof and conclude that the sequence as defined before converges to and.