Category Theory: Does the existence of $X\times (Y\times Z)$ imply the existence of $X\times Y$? Let $X\times Y$ be the product of objects $X$ and $Y$ in a category $C$, defined by the universal property of products. Is it true that the existence of the product $X\times (Y\times Z)$ implies the existence of $X\times Y$?

No. Consider the partial order $\mathbb{N}\cup\{\infty_1, \infty_2\}$ where $\infty_1$ and $\infty_2$ are incomparable to each other and greater than all natural numbers. Taken to be a category in the usual way (a unique morphism $x \to y$ if $x \le y$), products here are greatest lower bounds.
Then if $X = \infty_1$ and $Y = \infty_2$, the greatest lower bound of $X$ and $Y$ doesn't exist (the lower bounds are all natural numbers). However, if we take $Z = 0$, then $Y \times Z$ exists and equals $0$. Similarly, $X \times (Y \times Z)$ exists and equals $0$.
