According to these two links,( Link1 and Link2)
The distribution of multiplying two discrete random variables is given by :
\begin{equation*}
\begin{aligned}
Pr(T=t)&=\sum\limits_{x} Pr (X=x,Y=t/x)\\
&=\sum\limits_{x} Pr(Y=t/x|X=x) Pr(X=x) \\
&=\sum\limits_{x} Pr(X=x) Pr(Y=t/x) \text{ ( if X independent Y )}
\end{aligned}
\end{equation*}
or as :
\begin{equation*}
\begin{aligned}
Pr(T=t)&=\sum\limits_{x,y s.t. xy=t} Pr (X=x,Y=y)\\
&=\sum\limits_{x,y s.t. xy=t} Pr (X=x) Pr(Y=y) \text{ ( if X independent Y)}
\end{aligned}
\end{equation*}
Similarly for the sum of two discrete random variables ( Link3) is given by : \begin{equation*} % \scriptsize \begin{aligned} Pr(Z=z)&=\sum\limits_{x} Pr (X=x,Y=z-x)\\ &=\sum\limits_{x} Pr(Y=z-x|X=x) Pr (X=x) \\ &=\sum\limits_{x} Pr (X=x) Pr(Y=z-x) \text{ ( if X independent Y )} \end{aligned} \end{equation*}
How can I proof that this is correct ?