The phrase "applied to the proof" is correct and usable in written English.
It can be used in contexts where you are discussing methods, techniques, or principles that are relevant to a proof in mathematics, logic, or other fields.
Example: "The theorem can be applied to the proof of the conjecture, demonstrating its validity under certain conditions."
Alternatives: "relevant to the proof" or "pertinent to the proof".
The Galerkin method is applied to the proof.
For an acceleration applied to the proof mass, resistance change caused by beam deformation (Fig. 1a) is measured as voltage difference using external circuits.
We prove a general uniqueness theorem that can easily be applied to the proof of (generalized) Hyers-Ulam stability of the additive, quadratic, cubic, or the cubic-quadratic-additive type functional equation.
If a proof is found, answer-extraction techniques are applied to the proof to yield an answer [3], [4].
(2) Similar arguments to those given in the proof of (1) apply to the proof of assertion (2).
This metatheorem was then applied to the convergence proof of Halpern iterations by Shioji and Takahashi [24] for the extraction of rates of metastability in the setting of the metatheorem (though only relative to a given rate of metastability for the resolvent whose computation in this setting is still subject of ongoing research).
It also provides a product design example with multiple engineering changes applied to the implemented prototype system as proof of the implementation and the procedure.
These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation.
Proof Similar arguments can be applied to the three cases.
We observed that the chain rule can be applied to simplify the proof of Theorem 2.4.
Similar remarks apply to the demonstration that proofs, as defined nominalistically on p. 120, have the internal order of precedence among immediate sub-premisses and conclusions that we intuitively expect.
