Sentence examples for by the hypothesis there from inspiring English sources

By the hypothesis, there is an element U ∈ A such that ϕ ( U ) = I.

∀ ε > 0, by the hypothesis, there exists a suitable δ > 0 such that (1.8) holds.

For all ε > 0, by the hypothesis, there exists a suitable δ > 0 such that (2.1) holds.

Proof By the hypothesis, there exists an operator U ∈ ( A s ) 1 such that ϕ ( U ) = I.

For given ε > 0, by the hypothesis, there exists a δ = δ > 0 such that (4.10) holds.

(iv) By the hypothesis, there exists a nonempty compact and so a closed (resp., weakly closed and weakly compact) subset of and a point such that and (2.32)  .

If p ( x, T k + 1 x ) ≤ p ( x, T i x ) for some 0 ≤ i ≤ k, then by the induction hypothesis there holds p ( x, T k + 1 x ) ≤ p ( x, T i x ) ≤ M x.

As predicted by the second hypothesis, there were notable group differences in the speed of the brain responses and the number of discriminations between speech sounds.

If (H_0not subset E), then, by the induction hypothesis, there is a (u_0in {text {PSH}}_omega (H_0)) with ({Zin H_0: u_0(Z)=-infty }=Ecap H_0), and hence, by Lemma 3.4, there is a (v_0in {text {PSH}}_omega (mathbb {P}^n)) with ({Zin mathbb {P}^n: v_0(Z)=-infty }=Ecap H_0).

By the induction hypothesis, there is some j ≥ s k − 1 such that p ( T s k x, T i x ) ≤ { α p ( T s k − 1 x, T j x ), p ( T s k − 1 x, T s k − 1 x ) }.

By the hypothesis (H4), there exists a (N>0) such that (Vert xVert _{c}neq N).