In math, you provide a proof of something: a series of logical steps that lead to a conclusion. In the rest of life you provide proof of something: a citation or evidence or other support for a fact.
In the mathematical sense, the proof is somewhat divorced from the theorem it proves, in the sense that a single theorem could have many different proofs. (Most theorems are happy if they have one proof, but some of the more famous ones can be proved many different ways, and each of those ways is a proof.) A mathematical proof is also not a single-step thing: even the simplest proof will have at least three steps (premise→deduction→conclusion).
In the "real world" sense, proof is roughly synonymous with evidence. When you provide proof of insurance, you dig out a single piece of paper from your glove compartment; there's no logical deduction involved.
Bottom line is, your supervisor is correct: you provide a proof of Theorem A. What your English-major friend said isn't really true for math, and means something slightly different in any case.