Sentence examples for previous theorems given from inspiring English sources

Our results are improvement of the previous theorems given by Gao, Zheng and Chen.

The results of this paper are an extension of the previous theorems given by Chen and Chen and Qi.

Some results obtained in this paper improve and extend the previous theorems given by Gao, and five examples show the extension of solutions of the system of complex difference equation.

Some results of this paper improve and extend previous theorems given by Gao, and five examples are given to show the extension of solutions of the system of complex difference equations.

Our results of this paper are an improvement of the previous theorems given by Chen and Chen (Chinese Ann. Math., Ser. A 33(3):359-374, 2012) and Liu et al. (Appl. Math. J. Chin. Univ. Ser. B 27(1):94-104, 2012).

Applying the operator O b a x on both sides of (4.20) in a similar way as in the proofs of the previous theorems gives the result.

In this section, we present some results which show that several fixed point theorems on G-metric spaces are in fact direct consequences of the existence theorems given in the previous section.

These theorems give sharper estimates of (C_{p}(Omega)) than the previous estimates derived by the method in [19].

The Sylow theorems give a partial converse.

But more importantly, a comparison of the two theorems gives more insight in both algorithms.

The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Pythagorean theorem given above does not hold in a non-Euclidean geometry.