Sentence examples for proof propositions from inspiring English sources

Proof Propositions 5.1 and 5.2 indicate that T ˜ ( 1 / 2, q, t ) is increasing in q on [ − 1, 1 ] and T ˜ ( p, 1 − p, t ) is decreasing in p on [ 0, 1 / 2 ) and increasing on ( 1 / 2, 1 ]. It follows that T ˜ ( 1 2, 0, t ) < T ˜ ( 1 2, 1 2, t ) < T ˜ ( 3 4, 1 4, t ) < T ˜ ( 1, 0, t ), which, by some simplifications, yields the required inequalities.

Proof Proposition 5.1 verifies the axioms (A1)–(as) as well as efficiency  (4.6).

Arguing as in the proof of Proposition 3.1, Proposition 3.2 is proved under the following estimates.

We now proceed to prove Proposition 1. Proof of Proposition 1.

See the proof of Proposition 1 and Proposition 3 in Xiao and Yin [10].

The proofs of Proposition 1 and Proposition 2 are given in Proof of Proposition 1. Assume (A1) and (A2).

We end the proof by Propositions 1 and 6. □.

Before the proof of propositions, the following fact is necessary.

Proposition 3.4 and the proof of Proposition 4.1 is used to prove the following results.

Similar to the proof of Proposition 2.1, we can get Proposition 2.2.

Since the proof of Proposition 4 is similar to that of Proposition 3, we give only the proof of Proposition 3 below.