How to prove that $A × (B ∩ C) = (A × B) ∩ (A × C)$? I have to prove that $A × (B ∩ C) = (A × B) ∩ (A × C)$. While I know this is true by thinking about it I'm having a lot of trouble actually writing the proof. I'm relatively new to proofs so I have a lot of difficulty writing the equations that are necessary for the proof, all I really know how to do is write the whole thing out in words which isn't a very good proof.
 A: $$(a,b)\in A\times (B\cap C) \iff ((a\in A) \land (b\in B\cap C))\iff $$
$$ \iff ((a\in A) \land (b\in B) \land (b\in C)) \iff$$
$$ \iff (a,b) \in (A\cap B)\times (A\cap C)$$
A: The usual way to prove it is by "double inclusion" , i.e. :

$A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)$

and :

$(A \times B) \cap (A \times C) \subseteq A \times (B \cap C)$.

For the first part, assume $x \in A \times (B \cap C)$; this means that $x$ is an odered pair $(a,z)$ such that $a \in A$ and $z \in B \cap C$.
But $z \in B \cap C$ iff $z \in B$ and $z \in C$, and thus :

from $x \in A \times (B \cap C)$ we have : $x=(a,z)$ with : $a \in A, z \in B, z \in C$.

But now : $x=(a,z) \in A \times B$ and $x=(a,z) \in A \times C$ and so : $x \in (A \times B) \cap (A \times C)$.
Having showed that :

if $x \in A \times (B \cap C)$, then $x \in (A \times B) \cap (A \times C)$,

we can conclude with :

$A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)$.

The same for the other part of the proof.
