Sentence examples for proof given by from inspiring English sources
The proof given by Hiai and Petz is quite different.
According to the proof given by Chipot et al. [25], we have the following theorem.
The proof given by the authors is decisively based on the following result.
This leads us to the last mistake in the proof given by the authors of Theorem 3.3.
It is possible to prove this theorem by following the proof given by Branciari up to the point of showing that { f n ( x ) } is a Cauchy sequence for each x ∈ X.
The proof given by Kelisky and Rivlin is based on linear algebra tools, it involves the Stirling numbers of the second kind, and eigenvalues and eigenvectors of some matrices.
The manuscript, a set of 144 page proofs given by one of Hawthorne's relatives to the Natick Historical Society in Massachusetts in 1886, sold for a record price for a 19th-century literary work, the auction house said, eclipsing the $225,500 sale of an autographed manuscript of the 1823 poem "A Visit From St. Nicholas," also by Christie's, in 1994.
Our proofs are different from the proofs given by Elcrat [1].
In this paper, the proofs given by Malis are discussed and it will be shown that there is a limitation to the validity of the ESM.
On the one hand, the proofs given by the authors of the main results in [3] are incorrect because they are based on Lemma 3, which is false.
The nice proof we reproduce here is taken from the PhD Thesis of del Río [9], and is based on explicitly developing the idea underlying one of the two proofs given by Day.
