Proving Sets are Equivalent with Cartesian Products $A \times B \times (A \cap C) \subseteq (A \times B \times A) \cap (A \times B \times C)$
I need some help with proving this statement.  So far I've got:
If $x$ in an element in $A \times B \times (A \cap C)$, then $x$ is also an element in $(A \times B \times A) \cap (A \times B \times C)$ 
For $A \times B \times (A \cap C)$: $A \times B =$ {$a \in A,b\in B$} and for $A \cap C$= {$a \in A$ and $a \in B$}.
I'm probably wrong but the only thing I can come up with is because $a \in A$ in both cases $a \in A$ for $A \times B \times (A \cap C)$.    I haven't taken a look at the right side yet because I'd like to know how to do the left side properly.
 A: Any time you’re asked to prove a statement of the form $X\subseteq Y$, you should think first of the element-chasing approach: let $x$ be an arbitrary element of $X$, and show somehow that this forces $x$ to be an element of $Y$. If you can do that, you’ve proved that $X\subseteq Y$.
In your case that means starting out like this:

Let $x\in A\times B\times(A\cap C)$ be arbitrary.

Now use the definition of $A\times B\times(A\cap C)$ to see what this tells you about $x$. Right away it tells you that $x$ is an ordered triple whose first component is an element of $A$, whose second component is an element of $B$, and whose third component is an element of $A\cap C$.

Then there are $a\in A$, $b\in B$, and $c\in A\cap C$ such that $x=\langle a,b,c\rangle$.

Now you want to deduce that $x\in(A\times B\times A)\cap(A\times B\times C)$, which means showing that $x\in A\times B\times A$ and $x\in A\times B\times C$. Do you see how to do this?
A: Choose any $(x,y,z) \in A \times B \times (A \cap C)$ so that $x \in A$ and $y \in B$ and $z \in A \cap C$. 


*

*Then $z \in A$ and $z \in C$, so $(x,y,z) \in A \times B \times A$ and $(x,y,z) \in A \times B \times C$.

*But then $(x,y,z) \in (A \times B \times A) \cap (A \times B \times C)$, as desired.

