Answer
$
For\,\,all\,\,sets\,A, B, and\,\,C\\
if\,A\subseteq B\,\,then\,A\cap C\subseteq B\cap C \\
proof:\\
x\in A\cap C\ (by\,\,def.of\,\,inter\! section\,of\,sets)\\
x\in A\,\,and\,\,x\in c\,, \,\,\,\because A\subseteq B \\
x\in B\,\,and\,\,x\in c(by\,\,def.of\,\,inter\! section\,of\,sets) \\
\therefore x\in B\cap C\\
\therefore\,\,A\cap C\subseteq B\cap C
$
Work Step by Step
$
For\,\,all\,\,sets\,A, B, and\,\,C\\
if\,A\subseteq B\,\,then\,A\cap C\subseteq B\cap C \\
proof:\\
x\in A\cap C\ (by\,\,def.of\,\,inter\! section\,of\,sets)\\
x\in A\,\,and\,\,x\in c\,, \,\,\,\because A\subseteq B \\
x\in B\,\,and\,\,x\in c(by\,\,def.of\,\,inter\! section\,of\,sets) \\
\therefore x\in B\cap C\\
\therefore\,\,A\cap C\subseteq B\cap C
$