\[M \subset N \Longrightarrow M^{a} \subset N^{a}\]
証明
\begin{align*}
&A \subset B \Longrightarrow A^i \subset B^i \\
\Longleftrightarrow~ &A \subset B \Longrightarrow A^{cac} \subset B^{cac}\\
\Longleftrightarrow~ &A \subset B \Longrightarrow A^{ca} \supset B^{ca}
\end{align*}\(A,B\)をそれぞれ\(N^c,M^c\)にかきかえると
\begin{align*}
&N^c \subset M^c \Longrightarrow N^{cca} \supset M^{cca} \\
\Longleftrightarrow~ &M \subset N \Longrightarrow M^{a} \subset N^{a}
\end{align*}
証明終