By definition of product (Def. 3.81), there are morphisms\$$\pi_1:x\times y\to x\$$, \$$\pi_2:x\times y\to 2\$$. So, \$$x\leq(x\times y)\$$ and \$$y\leq(x\times y)\$$. This means that \$$x\times y\$$ is an upper bound for \$$x\$$ and \$$y\$$. By the universal property of products, any \$$Z\$$ such that there are morphisms \$$f:Z\to x\$$ and \$$g:Z\to y\$$ has a unique morphism into \$$x\times y\$$. Thus, \$$x\times y\$$ is a least upper bound for \$$x\$$ and \$$y\$$, i.e., it equals \$$x\wedge y\$$.