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Is there a Latin phrase that would be used when accepting some statement without providing the proof of such a statement?

For example, say you are working on an elementary number theory proof, and you make the statement "since $p$ is odd, $p^2$ is odd, which we accept [Latin phrase for 'without showing proof']." Obviously this is a simple thing to prove, but in some cases it might be nice to acknowledge that a proof exists and we do not wish to show it.

Could ex facie be used in such a situation ("we accept ex facie that $q$ odd implies $q^2$ odd")? Or failing the existence of a Latin phrase, is there a way that sounds a little less crude than "without proof"?

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Well, there is an English phrase. Why the need for Latin? – Qiaochu Yuan Aug 8 '12 at 20:34
Why do you want it to be Latin? It will be much more readable simply to write something like "Let's assume as given that such-and-such" or "It can be proved that such-and-such". – Henning Makholm Aug 8 '12 at 20:35
The request for a Latin phrase is because I feel as if I've encountered one once upon a time, and I don't recall where -- just that I had to ask someone what exactly it was. Also, the English phrasing, "we state without proof that $q$ odd implies $q^2$ odd" seems a little patronizing, as similar wording is used in elementary textbooks for when a concept is deemed to complicated for the reader. I'm looking for a phrase that could be used in place of the dreaded "clearly," except that we acknowledge that proof of the statement exists, rather than hoping it does. – Emily Aug 8 '12 at 20:54
"The proof is left as an exercise for the reader". – Robert Israel Aug 8 '12 at 21:16
Just do what the author of my linear programming textbook does and claim that everything is "easy to see". As a bonus, it's a good way to make readers feel bad about themselves. – crf Aug 8 '12 at 23:10
up vote 5 down vote accepted

It appears as though ex facie bears the intended meaning. Prima facie has a similar meaning.

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Arguendo is close, and might better fit proofs by contradiction, or derivations from a conjecture.

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Are you thinking of a priori?

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This is not an answer - you should post it as a comment. – Dennis Gulko Oct 15 '13 at 14:56
No. A priori means before the fact, which is different. The desired term instead would communicate that we're accepting a result on face value, without detailed examination that the result is true, perhaps because exploration would be a distraction from the intended result, or alternatively that demonstrating the result would be too lengthy. $${}{}$$As an example, consider writing a clever homework solution in an analysis class that uses, say, the Sylow theorems. You wouldn't want to prove them, but you might want to use them. – Emily Oct 16 '13 at 16:42
@DennisGulko look at the poster's reputation, it takes 50 reputation to post comments. – Olivier Bégassat Dec 10 '13 at 2:45

Axiom (something that is accepted without proof) ex.: this statement is an Axiom.
Axiomatic, ex.: this statement is axiomatic in every sense.
Axiomatically, ex.: this statement is axiomatically accepted by some.
Note: Of Greek origin but surely used in Latin as well.

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