Proof of the chain rule

Available under Creative Commons-ShareAlike 4.0 International License.

In the statement of the chain rule on page 37, I followed my usual custom of writing derivatives as dy/dx, when actually the derivative is the standard part, st(dy/dx). In more rigorous notation, the chain rule should be stated like this:

The transfer principle allows us to rewrite the left-hand side as st[(dz/dy)(dy/dx)], and then we can get the desired result using the identity st(ab) = st(a)st(b).