Show that for positive integers x, y , z and w If x < y and z < w then zx < wy

Question

My attempt at solving this goes $$z\lt w\Rightarrow z\bullet z\lt w\bullet w$$ since $z\lt w$ is logically equivalent to $x\lt y$ just with different symbols, then $$z\lt w \Rightarrow z\bullet x \lt w\bullet y$$ However, in the proof of the first theorem, I took for granted that $w\bullet z = z\bullet w$ which is not true in the second theorem. I am out of ideas, any help would be appreciated. Thank you!