Polynomial calculus is a proof system with two derivation rules: $${p ~~~~ q\over \alpha p + \beta q} ~~~~~~ {p \over x_i \cdot p}~~~~~~ {\over x_i^2 - x_i}$$ where $p,q \in \mathbb{F}[x_1, \ldots, x_n]$, $\alpha, \beta \in \mathbb{F}$, $\mathbb{F} $ is a field.
My lecture note says that for every multilineal $f$ one can derive $f\big|_{x_i=0} \cdot (1-x_i)$ and $f\big|_{x_i=1} \cdot x_i$ from $f$ but I've tried to do so and failed, all useful polynomial I can derive from $f$ are $f x_i$ and $(1-x_i) f$ and I don't see how to derive the needed polynomials from them. Any ideas?
Well, turned out that it's simple: for every multilinear $f = x_i f\big|_{x_i=1} + (1-x_i) f\big|_{x_i=0}$, therefore $(1-x_i) f\big|_{x_i=0} = f - x_i(1-x_i) (-f\big|_{x_i=1} + f\big|_{x_i=0})$. The latter could be derived from the axiom $x_i^2 - x_i$.
$x_i f\big|_{x_i=1}$ could be derived in a similar way.