term in $f$.

If $\sigma_1 \subseteq \sigma$ but $\sigma_2 \not \subseteq \sigma$, then $x_\sigma$ is not a term in $p_c$ for any $c \in C_2$ or for any $c \in C_1$ such that $c \not \subseteq \sigma$. But $x_\sigma$ appears once as a term in $p_c$ for all $c \in C_1$ such that $c \subseteq \sigma$. There are $2^m$ such $c$, where $m = |\sigma \setminus \sigma_1|$. Therefore, if $m \geq 1$, $x_\sigma$ will be added an even number of times in the expansion of $f$ and will therefore cancel out since we are working in $\mathbb{F}_2[x]$. We get $m=0$ only when $\sigma = \sigma_1$, which tells us that $x_{\sigma_1}$ is a monomial term in $f$. Similar reasoning applies when $\sigma_2 \subseteq \sigma$ but $\sigma_1 \not \subseteq \sigma$.

Finally, if $\sigma_1 \cup \sigma_2 \subseteq \sigma$, then $x_\sigma$ is a term in $p_c$ for any $c \in C_1 \cup C_2$ such that $c \subseteq \sigma$. As before, we need to determine how many $c \in C_1 \cup C_2$ are contained in $\sigma$. We know there are $2^m$ codewords $c$ such that $\sigma_1 \subseteq c \subseteq \sigma$ and $2^k$ codewords $c'$ such that $\sigma_2 \subseteq c' \subseteq \sigma$, where $m = |\sigma 