Sentence examples for a concise proof from inspiring English sources

Proof We give only a concise proof.

Proof We give again only a concise proof.

As we use the eigenvalues in the sequel, we present a concise proof.

A concise proof of the First Theorem can also be found in [38].

In Section 4, we give a concise proof for an L p extension of Theorem 1 of [15].

In the following, we will give a concise proof for the multiplicity of T-periodic solutions independent of Theorem 4.2.

A better and concise proof of a fundamental proposition concerning the refracted flux across a bi-material interface in the conductivity problem is also provided.

gThis concise proof is based on Uzawa's proof.

Their formulation and presentation of the trust (lambda )-calculus adopt a small-step semantics and a Church style type system, supporting concise proofs of type soundness, erasure and simulation theorems.

This allowed us to give concise proofs of type soundness, erasure and simulation theorems.

This section presents concrete mathematical analysis results on unique existence and comparative statics on τ * with concise proofs.