TITLE: What does a proof in an internal logic actually look like?
QUESTION [39 upvotes]: The nLab has a lot of nice things to say about how you can use the internal logic of various kinds of categories to prove interesting statements using more or less ordinary mathematical reasoning. However, I can't find a single example on the nLab of what such a proof actually looks like. (The nLab has a frustrating lack of examples in general.)
Can anyone supply me with some examples? I'd be particularly interested in the following kinds of examples:

I've heard in a topos one can internalize the real numbers and, in the topos $\text{Sh}(X)$ of sheaves on a topological space, this reproduces the sheaf of continuous real-valued functions $X \to \mathbb{R}$. Moreover, one can internalize "finitely generated projective $\mathbb{R}$-module" and in $\text{Sh}(X)$ this reproduces real vector bundles on $X$. What can you prove about vector bundles this way? 
I'd also like to see examples of what you can do in the internal logic of Cartesian closed categories. 

This MO question is related but it doesn't really satisfy my curiosity.

REPLY [4 votes]: You can find examples in the work by Christopher Mulvey, for example

Intuitionistic algebra and representations of rings. In Recent advances in the representation theory of rings and C*-algebras. Mem. Amer. Math. Soc., 148, 3-57 (1974).

develops (essentially) topos theory and its internal logic from scratch and discusses the example of vector bundles.