We will prove this true by the definition of set equality.
Suppose that \( \left ( x , y \right ) \in \left ( A \times B \right ) \cap \left ( C \times D \right ) \), which is true iff \( \left ( x , y \right ) \in \left ( A \times B \right ) \), so that \( x \in A \) and \( y \in B \).
We also know that \( \left ( x , y \right ) \in \left ( C \times D \right ) \) which is equivalent to \( x \in C \) and \( y \in D \)
We have \( x \in A \cap C \) and \( y \in B \cap D \) which is equivalent to \( \left ( x , y \right ) \in \left ( A \cap C \right ) \times \left ( B \cap D \right ) \).
Thus we've proven that \( \left ( x , y \right ) \in \left ( A \times B \right ) \cap \left ( C \times D \right ) \) if and only if \( \left ( x , y \right ) \in \left ( A \times B \right ) \cap \left ( C \times D \right ) \) as needed.
So suppose that \( \left ( x , y \right ) \in \left ( A \cap B \right ) \times C \), which is true iff \( x \in A \cap B \) and \( y \in C \), which is true iff \( \left ( x , y \right ) \in A \times C \) and \( \left ( x , y \right ) \in B \times C \), which is true iff \( x \in \left ( A \times C \right ) \cap \left ( B \times C \right ) \).
Therefore \( \left ( x , y \right ) \in \left ( A \cap B \right ) \times C \) if and only if \( x \in \left ( A \times C \right ) \cap \left ( B \times C \right ) \), so \( \left ( A \cap B \right ) \times C = \left ( A \times C \right ) \cap \left ( B \times C \right ) \) as needed.