Sentence examples for as defined in Theorem from inspiring English sources

We take as defined in Theorem 2.1.

Theorem 2 D. Let, be as defined in Theorem A. Define and for and, where are natural numbers.

Theorem 2 B. Let, be as defined in Theorem A. Let and be positive sequences for and where are natural numbers.

Proof We have y n = ( 1 − β n ) x n + β n T x n + v n, x n + 1 = ( 1 − α n ) x n + α n T y n + u n, n ≥ 0. Let x 0 ∈ K be as defined in Theorem G1 with x 0 ≠ x ∗.

and are as defined in Theorem 2.1.

Let be as defined in Theorem 3.1.

The following consequence of Theorem 1 points out another connection between the presentation in (6) as defined in either Theorem 1 or Corollary 1 and generating functions.

where is as defined as in Theorem 2.5.

Let be a Borel set satisfying Corollary 1.2 (ii), defined as in Theorem 1.1, and defined by (1.23).

Assume that, and are defined as in Theorem 3.1, is defined as in Lemma 2.2 such that for with ; then (3.26).

Let be defined as in Theorem 1.1 and defined by (1.23).