In expressing $B(x,y)$ as an infinite product using the infinite product definition of $\Gamma(x)$, the book Special Functions by Andrews, et al doing this way : $$B(x,y)= \dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} = \dfrac{(x+y)e^{\gamma (x+y)} \prod_{n=1}^{\infty}(1+\frac{x+y}{n})e^{-\frac{x+y}{n}}}{xe^{\gamma x} \prod_{n=1}^{\infty}(1+\frac{x}{n})e^{-\frac{x}{n}}ye^{\gamma y} \prod_{n=1}^{\infty}(1+\frac{y}{n})e^{-\frac{y}{n}}} = \dfrac{(x+y)}{xy} \prod_{n=1}^{\infty} \dfrac{(1+\frac{x+y}{n})}{(1+\frac{x}{n})(1+\frac{y}{n})}.$$ The last inequality is true if the following is true : $$\dfrac{\prod_{n=1}^{\infty}(1+\frac{x+y}{n})e^{-\frac{x+y}{n}}}{\prod_{n=1}^{\infty}(1+\frac{x}{n})e^{-\frac{x}{n}}\prod_{n=1}^{\infty}(1+\frac{y}{n})e^{-\frac{y}{n}}} = \prod_{n=1}^{\infty} \dfrac{(1+\frac{x+y}{n})}{(1+\frac{x}{n})(1+\frac{y}{n})} \ \ (*) \ .$$
My question : How do you justify $(*)$, i.e. how do you collect terms of each infinite product such a way that you select nth from each, each time?
I have studied theorems in reordering infinite sums but not on infinite products.