Sentence examples for proofs is given from inspiring English sources

For linear logic, this succinct representation of proofs is given by proof nets, graph-like structures that enjoy particularly good properties for the MLL fragment of the logic.

A complete outline of Reproducing Kernel Hilbert space (RKHS) and the derivation from 3 stated local minimax results (to which the reader may give his/her own proofs) is given in the appendix.

Convergence proofs are given in the Appendix.

Related proofs are given in Section 4.

Different proofs were given by Pisier [7].

The proofs are given in [3].

The rigorous proofs are given by using Lyapunov theory.

First, some useful theorems and their proofs are given.

Two strict proofs were given in [22] and [26].

The detailed proofs are given in Section 3.

Auxiliary lemmas with proofs are given in Section 3.