Natural Number Multiplication is Cancellable for Ordering
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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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