I am currently reading Section 10 in *Rational Homotopy Theory* by Felix, Halperin and Thomas.

A rather important step after the construction of the simplicial cochain algebra $A = A_{PL}$ is proving they are extendable, meaning that for any (not necessarily strict) subset $I$ of $\{0,\ldots,n\}$ and simplices $\Phi_i$ for which $\partial_i\Phi_j = \partial_{j-1}\Phi_i$, $i<j$ in $I$, there is $\Phi$ such that $\partial_i \Phi = \Phi_i$.

This is claim $(iii)$ in Lemma 10.7, page 123 of the first edition.

Concretely, $A_n$ is the free commutative DG-algebra generated by $t_0,\ldots,t_n$ in degree $0$, by $y_0,\ldots,y_n$ in degree $1$ and with $dt_i = y_i$, modulo the relations that $\sum t_i = 1$ and $\sum y_i=0$, face/coface maps given by thinking about the $t_i$ as the barycentric coordinates of the $n$-simplex. Equivalently, it is generated by $t_1,\ldots,t_n$ and $y_1,\ldots,y_n$ subject to commutativity only and $t_i$ covers $y_i$.

I find the proof really obscure; although with a bit of work I can follow it, I cannot get anything out of it. I was wondering if someone could shed some light on the proof, or else point to other proofs (if there are any). I admit the whole book is rather terse at moments, but in general, independent of this, I can gather some intuition of what is going on, but this particular bit is eluding me.