How to show that $\prod_{i=1}^n a_i = \prod_{i=1}^{r} a_i \times \prod_{i=r+1}^{n} a_i$?

Question

I need to prove this multiplication property. Is it correct?

Suppose that $1\le r\le n$.

Show that:

$$\prod_{i=1}^n a_i = \prod_{i=1}^{r} a_i \times \prod_{i=r+1}^{n} a_i. $$

Answer 1

I assume we are working in a context where multiplication is associative, so $a\times(b\times c) = (a\times b)\times c$

Then it is clear, since $$ \prod_{i=1}^{n} a_i = a_1 \times ... \times a_r \times a_{r+1} \times ... \times a_n = (a_1 \times ... \times a_r ) \times (a_{r+1} \times ... \times a_n ) = \prod_{i=1}^{r} a_i \times \prod_{i=r+1}^{n} a_i$$