This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. In particular, the goal of the paper is to attempt to provide a rigorous analysis of these explanations. This will be done into two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof-trees together with a decomposition of their elements into assumptions, conclusion and rules; secondly we will show that those mathematical proofs that are regarded as having an explanatory power all have the same formal presentation: they display an increase of conceptual complexity from the assumptions to the conclusion.