When you consider the title of this play, proof has two significant meanings. First, there is the more common use of the word, where proof refers to evidence which establishes the truth or validity of a claim or concept. This definition of proof is relevant to this story because the main characters spend much of the play trying to prove who wrote the mathematical proof the Hal discovers.
This brings us to the second relevant meaning of proof. Here, the title refers to the mathematical definition of the word. In math, proof refers to a series of mathematical statements that posit that a particular conclusion is demonstrably and consistently true. In the play, Hal and Catherine are searching for two proofs. One is the mathematical proof that was found in the professor's desk and the other is proof of who wrote it.