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The phrase "a satisfactory proof" is correct and usable in written English.
It can be used when discussing evidence or arguments that adequately support a claim or conclusion.
Example: "The researcher presented a satisfactory proof of the theory, convincing the committee of its validity."
Alternatives: "an adequate demonstration" or "a convincing argument".
Exact(2)
Metaphysics stands or falls with the solution of this problem, or on a satisfactory proof that the possibility it requires to be explained does not in fact obtain".
A satisfactory proof of the equivalence of wave mechanics and matrix mechanics was only provided later, when von Neumann (1932) showed that both approaches could be seen as different realisations of an abstract Hilbert space formalism.
Similar(57)
Arizona, however, adopted a state law requiring "satisfactory proof" of U.S. citizenship to register, such as a birth certificate, U.S. passport, or state driver's license that shows citizenship.
The difficulty in proving the reliability and durability of a bonded repair is also identified and suggestions for a quality management system proposed to provide satisfactory proof of compliance to design standard requirements.
You also pride yourself as an independent thinker, and do not accept others' statements without satisfactory proof.
("You pride yourself as an independent thinker and do not accept others' statements without satisfactory proof". "While you have some personality weaknesses, you are generally able to compensate for them").
But, by law, you must provide satisfactory proof of your identity.
Those restrictions on sovereignty are usually explained as deriving from consent or autolimitation, but it can easily be demonstrated that in some cases states have been considered as bound by certain rules of international law despite the lack of satisfactory proof that these rules were expressly or implicitly accepted by them.
However, no satisfactory proof of it has been given.
Finally Dedekind indicates how, along such lines, explicit and straightforward proofs of various facts about the real numbers can be given, including ones that had been accepted without satisfactory proof so far.
Two out of five plaintiffs weren't able to offer satisfactory proof of the stipulated security measures, so their suits were dismissed.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com