The phrase "and proved in a" is correct and usable in written English.
It can be used when discussing evidence or validation of a claim, theory, or hypothesis in a specific context.
Example: "The results of the experiment were significant and proved in a controlled environment."
Alternatives: "and demonstrated in a" or "and validated in a".
The model is tested and proved in a cassava starch factory in Tayninh, Vietnam.
The character of the solution of the eigenvalue problem of the constrained system is stated and proved in a Theorem.
Three theorems are stated and proved in a sequence in support of the HLEFS ability to keep the output error in a relatively small bound.
In this paper, we present a generalization of a combinatorial lemma we stated and proved in a recent work.
A non-equivalent but sufficient linearization form of IJLCP was proposed and proved in a straightforward manner for solving this type of problem.
Reich [9] extended Browder's result to the setting of Banach spaces and proved, in a uniformly smooth Banach space, that x t converges strongly to a fixed point of T and the limit defines the (unique) sunny nonexpansive retraction from C onto F ( T ).
This was tested and proved in an experiment in 9 Goettingen mini-pigs with elevation heights of 5, 10 and 15 mm.
In this way, small but significant metal recovery gains may be measured and proven in a plant trial.
For the proof of these Theorems we need a lemma of independent interest, which we state and prove in a little more general form.
A simple and natural way to unify and prove in a simple manner several metrical fixed point theorems is to consider an implicit contraction type condition instead of the usual explicit contractive conditions.
and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for groups.
