Sentence examples for proof below from inspiring English sources

The proof below is a modification of the proof of Theorem 1.3.

The proof below is exactly the same as that of Lemma 2.2, except that the coefficients involved are different and hence Lemma 3.6 is essential.

To simplify the notation, (l sigma):=l(V sigma ))), (sigmain T), or σ is a face of a simplex in T. The proof below is an adaptation of the proof of Lemma 1 in [1].

In this section, we shall prove Theorem 2. As we will see in the proof below, we need only to consider the case that ∂ 3 ( u + b ) ∈ L 2 1 − r ( 0, T ; X ˙ r ) ( 0 ≤ r ≤ 1 ).

In the proof below, we see that 0 < t 1 < 1 if 0 < θ < 1. Proposition 3.3 shows that M K ( θ ; a, b ) is better than M A in a neighborhood of the diagonal a = b in the quarter plane ( 0, ∞ ) × ( 0, ∞ ).

As proof, below a streaming news ticker ("Gay Basher Gets 12 Years") are panels about "Confronting Hate in America": Two Latinos are beaten on Long Island; a white supremacist shoots Jews in Los Angeles; a Sikh is murdered in a post-9/11 "hate crime"; a homosexual student is brutally murdered in Wyoming.

We mention underlying two product graphs G= V,E) with V=V k+1; in the proofs below, an extended Markov chain (cf. [11]) will moves on the graphs.

For the completeness of the paper, we give proofs below.

The proofs below also require the following lemma.

We now give some basic definitions and theorems concerning an iterated function system, which are used in the proofs below.

The hypersurfaces and are easier to work with, and we utilize them instead of S π and in our proofs below.