Natural Number Multiplication is Cancellable for Ordering

Theorem

Let $\N$ be the natural numbers.

Let $\times$ be multiplication on $\N$.

Let $<$ be the strict ordering on $\N$.

Then:

$\forall a, b, c \in \N: a \times c < b \times c \implies a < b$

$\forall a, b, c \in \N: a \times b < a \times c \implies b < c$

That is, $\times$ is cancellable on $\N$ for $<$.

Proof

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