Define $\displaystyle \phi: A$ x $\displaystyle B$ ---> $\displaystyle A/C$ x $\displaystyle B/D$ by $\displaystyle \phi((a,b))=(aC,bD)$

I have to show this map is well-defined, here's what I did:

Let $\displaystyle (a_1, b_1), (a_2, b_2) \in A$ x $\displaystyle B$, and let $\displaystyle \phi(a_1,b_1)=\phi(a_2,b_2)$. Then $\displaystyle (a_1C,b_1D)=(a_2C,b_2D) \implies (a_1, b_1)=(a_2, b_2)$. Is this implication clear?

Can I write $\displaystyle (a_1C, b_1D) = (a_1, b_1)(C$ x $\displaystyle D)$?

Please hlep, thank you.