Sentence examples for this theorem is completed from inspiring English sources

The proof of this theorem is completed.

end{aligned} (3.18) The proof of this theorem is completed.

Hence, the proof of this theorem is completed.

By the argument of contradiction, the proof of this theorem is completed.

(17) By Applying Theorem A and (17), the proof of this theorem is completed immediately.

} Therefore, liminf_{trightarrowinfty}frac{log S t)}{t}geqfrac{r}{4}, which is a contradiction, and the proof of this theorem is completed.

Thus, the theorem is completed.

end{aligned} The proof of the theorem is completed.

Thus the proof of the theorem is completed.

(31) Hence the proof of the theorem is completed.

Therefore, the proof of the theorem is completed.