# Ordering on Natural Numbers is Compatible with Multiplication/Corollary

## Corollary to Ordering on Natural Numbers is Compatible with Multiplication

Let $a, b, c, d \in \N$ where $\N$ is the set of natural numbers.

Then:

$a > b, c > d \implies a c > b d$

## Proof

 $\displaystyle a$ $>$ $\displaystyle b$ $\text {(1)}: \quad$ $\displaystyle \leadsto \ \$ $\displaystyle a c$ $>$ $\displaystyle b c$ Ordering on Natural Numbers is Compatible with Multiplication

 $\displaystyle c$ $>$ $\displaystyle d$ $\text {(2)}: \quad$ $\displaystyle \leadsto \ \$ $\displaystyle b c$ $>$ $\displaystyle b d$ Ordering on Natural Numbers is Compatible with Multiplication

Finally:

 $\displaystyle a c$ $>$ $\displaystyle b c$ from $(1)$ $\displaystyle b c$ $>$ $\displaystyle b d$ from $(2)$ $\displaystyle \leadsto \ \$ $\displaystyle a c$ $>$ $\displaystyle b d$ Ordering on Natural Numbers is Trichotomy

$\blacksquare$