Affirmanti, non neganti, incumbit probatio

AFFIRMANTI, NON NEGANTI, INCUMBIT PROBATIO
a more common form: Ei incumbit probatio qui dicit, non qui negat

Meaning: The proof lies upon one who affirms, not upon one who denies.

The maxim provides a rule in establishing a presumption of who initially carries the burden of proof. This is a deep-rooted and well known maxim, mentioned in the Blackstone's Commentaries (Book 3, p.366 (1791)) as one established both in common and civil laws. It appears to be a part of the Justinian I's Corpus Juris Civilis (Digest book*), completed in 533 A.D. And even there, it is referred as "A maxim of Paulus," enforced by the Emperor Pius' rescript. Its full cite and a detailed account are given here.

Allegata Et Probata

ALLEGATA ET PROBATA 

Meaning: The things alleged must agree with the things proved.

This rule applies to the proofs presented at trial, holding that such proofs have to be corresponding with what was alleged in the pleadings. Proving a fact not alleged is disallowed as a "variance."
The rule believed to be applied strictly in the Medieval period, but got relaxed in more recent times, so that the presented proofs need only to relate to the allegations. Variances to some immaterial degree become allowed.