The phrase "adapted to prove" is correct and usable in written English.
It can be used when discussing something that has been modified or tailored for the purpose of demonstrating or establishing a point or fact.
Example: "The experiment was adapted to prove the hypothesis that temperature affects the rate of chemical reactions."
Alternatives: "designed to demonstrate" or "modified to establish".
Techniques used to ensure that airplanes won't stall out in flight could be adapted to prove that computer programs won't divide by zero.
In fact the games used for model-theoretic forcing (see the entry on logic and games) can be adapted to prove the omitting types theorem too.
For our purposes in this paper, the importance of Peano's original ideas is that they can be adapted to prove existence results for (1.1) under such weak conditions that standard functional analysis arguments are no longer valid.
Bearing in mind that a module M is quasi-injective if and only if it is invariant under any endomorphism of its injective envelope, the above arguments can be easily adapted to prove: Let R be a commutative noetherian ring and M, an R-module.
Finally, the same kind of arguments could be adapted to prove existence and uniqueness of an optimal map for other costs, in particular to costs of the form (c x,y)=h x-y)) for a strictly convex function (h:mathbb R^drightarrow mathbb R), which includes all the costs of the form (c(x,y)=h x-y^p) with (p>1).
Moreover, the limit theorems of Kurtz [48] can be adapted to prove convergence of solutions of the hybrid model to solutions of a corresponding Langevin approximation in the limit N → ∞ and finite time, where N is the number of ion channels within the membrane [44, 52].
It will be shown that previously developed graph theory algorithms for transmission path analysis (TPA) in SEA can be adapted to SmEdA and prove useful for that purpose.
The kitten (a boy we've named "Pete" after the children's book character) very quickly adapted to domestic life, proving to be not only friendly but affectionate.
The full length version of the CPQ11 14 cross-culturally adapted to Brazilian Portuguese proved valid and reliable for its use on Brazilian children [ 15].
Several key tools and principles, adapted to health care, have proved effective in improving hospital operations.
"My suspicion is that they will prove well adapted to the special brand of artificiality that is reality television".
