Sentence examples for by the preceding theorem from inspiring English sources

By the preceding theorem, C ( I, T ) = F ( T ) is a nonempty set.

By the preceding theorem and Theorem 2.3, we obtain the following result.

According to the minimization principle which is given by the preceding Theorem 5.3 we can find the nth eigen-pair ((lambda_{n},phi_{n})) only after the previous eigenfunctions (phi_{1}(x),phi_{2}(x),ldots, phi_{n-1}(x)) are known.

Next, since ∀ p ∈ F x ∃ q ∈ F y ; M ( p, q, t ) ≥ 1 − α + α M ( x, y, t ) ≥ 1 − α > 0, for all comparable elements x, y ∈ X and t > 0, there exist x 0 ∈ Y and x 1 ∈ F x 0 such that M ( x 0, x 1, 0 + ) > 0. Consequently, by the preceding theorem, F has a fixed point.

Furthermore, by virtue of the preceding theorem F_{n} phi_{1},ldots2},ldots, phi_{n-1})=lambda_{n}.

Theorem (BF): ∀x□φ → □∀xφ Proof: Corollary (BF): ◇∃xφ → ∃x◇φ Proof: Immediate from the preceding theorem, by propositional logic and the definitions of '∃' and '◇'.

Let be as in the preceding Theorem 4.11.

The following theorem shows that the converse of the preceding theorem does not hold.

In view of Theorem 3.1, we have the following corollary of the preceding theorem.

We shall establish fixed point theorems which improve in several directions the preceding theorem.

Note that, from the preceding theorem, one cannot deduce (L^{p}(mathbb {R}^{n};v)) boundedness for the commutator operator by taking (lambda=-1/p), just in the case of Theorem 3.3.