Sentence examples for fundamental proof of from inspiring English sources

Although one hybrid line showed phenotypic instability, our results provide a fundamental proof of principle, demonstrating that apomixis can indeed be used in plant breeding and seed production to fix complex, quantitative phenotypes across generations.

Indeed, the enduring popularity of the argument from miracles underscores the truth of Butler's observation that miracles are one of the "direct and fundamental proofs" of Christianity.

They will also believe it in the face of irrefutable counter-evidence and millennia of fundamental proof.

The following lemma is fundamental in proofs of our main results.

This paper describes fundamental concepts and the proof-of-concept experiments, modeling and design, microfabrication processes, and initial testing results for the microfluidic devices based on the EW and EWOD principles.

The parallel postulate is fundamental for the proof of the theorem that the sum of the angles of a triangle is always 180 degrees.

The following lemmas are fundamental in the proof of our main results of this section.

The following fixed-point theorem is fundamental for the proof of the best approximation theorem.

The following Schaefer's fixed-point theorem is fundamental in the proof of our main results.

In the following, we present the useful lemmas which are fundamental in the proof of our main results.

Then there exists a constant c depending only on γ and θ such that for every ρ, R, R 0 ≤ ρ < R ≤ R 1, we have f ≤ c [ A ( R − ρ ) − γ + B ]. Now we present a very useful lemma which is fundamental in the proof of our theorems, it appears in [1] as follows.