In this proof of the Borsuk-Ulam theorem, it is said on page 5 that the 'Stack of Records Theorem' will be used:
The idea is that, given any smooth map $f:S^1\rightarrow S^1$, we can lift f locally using the Stack of Records Theorem and then patch the pieces together (...)
However, I'm having trouble to find where exactly is it needed during the following steps. I can see that the relationship between winding numbers and boundaries stated on page 3 is used on page 6 to conclude that $$\deg_{2}(f)=q=1 \mod2$$ from the fact that $$p(g(t+1/2))=p(g(t)+q/2),$$ and the 'Stack of Records Theorem' was indeed needed to prove that relationship, but I'm not sure if that's the step the quote is referring to.
Edit: Here's the statement of the 'Stack of Records Theorem' (according to Pollack's 'Differential Topology' book):
Suppose that y is a regular value of $f: X \rightarrow Y$, where $X$ is compact and has the same dimension as $Y$. Then $f^{-1}(y)$ is a finite set ${x_1,\ldots,x_n}$, and there exists a neighborhood $U$ of $y$ in $Y$ such that $f^{-1}(U)$ is a disjoint union $V_{1}\cup\ldots\cup V_{n}$ where $V_i$ is an open neighborhood of $X_i$ and $f$ maps each $V_i$ diffeomorphically onto $U$.