$C'$) as follows: $$\label{eqn:U_prime}
       U_i' = 
       \begin{cases}
            U_i & \text{if $i \not \in \tau_1 \cup \tau_2$} \\
            U_i^c & \text{if $i \in \tau_1 \cup \tau_2$.} \\
        \end{cases}$$

Because $\sigma_1 \cup \sigma_2$ and $\tau_1 \cup \tau_2$ are disjoint and $U_{\sigma_1} \cap (\cap_{i \in \tau_1} U_i^c) = U_{\sigma_2} \cap (\cap_{j \in \tau_2} U_j^c) $, we know that $U_{\sigma_1} '\cap (\cap_{i \in \tau_1} U_i') = U_{\sigma_2}' \cap (\cap_{j \in \tau_2} U_j')$ as well. Therefore, we can say from the first case that $x_{\sigma_1 \cup \tau_1} + x_{\sigma_2 \cup \tau_2} \in J_{C'}$.

Now we need to show that the corresponding pseudo-monomial $x_{\sigma_1} \prod_{i \in \tau_1}(1+x_i) + x_{\sigma_2}\prod_{j \in \tau_2}(1+x_j)$ is in $J_C$. Consider the $i$-th [*bit flip map*]{} $\delta_i$ : $\mathbb{F}_2[x_1,...,x_n] \rightarrow \mathbb{F}_2[x_1,...,x_n]$ given by

$$\delta_i (x_j) = 
       \begin{cases}
            x_j & \text{if $j \neq i$} \\
            1+x_j & \text{if $i=j$} \\
        \end{cases}$$

as defined in Jeffs, Omar, and Youngs [@Neural_homo], and let $\delta_{\tau_1 \cup \tau_2}$ be the composition of all $\delta_i$ such that $i \in \tau_1 \cup \tau_2$. Also as in [@Neural_homo], for a neural code $C$ we define the following code: $$\delta_i(C) := \{ u \in \mathbb{F}_2^n \mid supp(u) = supp(c) \oplus  \{i\} \text{ for some } c \in C\},$$ where $\oplus$ denotes 