Let $A, B$ be groups and $C,D$ be normal subgroups of $A,B$ respectively. The question was proving $A/C\times B/D\cong (A\times B)/(C\times D)$; which can be solved with 1st homomorphism theorem.
However, I wonder $A/C\times B/D = (A\times B)/(C\times D)$.
My attempt is:
\begin{align} (A\times B)/(C\times D)=\{(a,b)(C\times D)\ | \ a\in A, b\in B\} \\ = \{(a,b)(C , D)\ | \ a\in A, b\in B\} \\ =\{(aC,bD) \ | \ a\in A, b\in B\} \\ = A/C\times B/D, \end{align} since $(a,b)(C\times D)=\{(a,b)(c,d) \ | \ c\in C,d\in D\}=\{(x,y) \ | \ x=ac, y=bd, c\in C, d\in D\}$
$=\{(x,y) \ | \ x\in aC, y\in bD\}=(aC,bD)$.
I think it is wrong, but I can't find wrong point in my attempt above, and can't think of any counterexample. Are there any errors in my attempt? Or can you please explain a counterexample of $A/C\times B/D = (A\times B)/(C\times D)$? Any help will be greatly appreciated.