Natural Number Addition is Cancellable for Ordering

Theorem
Let $\N_{> 0}$ be the 1-based natural numbers:
 * $\N_{> 0} = \left\{{1, 2, 3, \ldots}\right\}$

Let $<$ be the (strict) ordering on $\N_{> 0}$ defined as Ordering on Natural Numbers:
 * $\forall a, b \in \N_{>0}: a < b \iff \exists c \in \N_{>0}: a + c = b$

Then:
 * $\forall a, b, c \in \N_{>0}: a + c < b + c \implies a < b$
 * $\forall a, b, c \in \N_{>0}: a + b < a + c \implies b < c$

That is, $+$ is cancellable on $\N_{>0}$ for $<$.

Proof
By Ordering on Natural Numbers is Trichotomy, one and only one of the following holds:
 * $a = b$
 * $a < b$
 * $b < a$

Let $a + c < b + c$.

Suppose $a = b$.

Then by Ordering on Natural Numbers Compatible with Addition:
 * $a + c = b + c$

By Ordering on Natural Numbers is Trichotomy, this contradicts the fact that $a + c < b + c$.

Similarly, suppose $b < a$.

Then by Ordering on Natural Numbers Compatible with Addition:
 * $b + c < a + c$

By Ordering on Natural Numbers is Trichotomy, this also contradicts the fact that $a + c < b + c$.

The only other possibility is that $a < b$.

So
 * $\forall a, b, c \in \N_{>0}: a + c = b + c \implies a < b$

and so $+$ is right cancellable on $\N_{>0}$ for $<$.

Let $a + b < a + c$.

Suppose $b = c$.

Then by Ordering on Natural Numbers Compatible with Addition:
 * $a + b = a + c$

By Ordering on Natural Numbers is Trichotomy, this contradicts the fact that $a + b < a + c$.

Similarly, suppose $c < b$.

Then by Ordering on Natural Numbers Compatible with Addition:
 * $a + c < a + b$

By Ordering on Natural Numbers is Trichotomy, this also contradicts the fact that $a + b < a + c$.

The only other possibility is that $b < c$.

So
 * $\forall a, b, c \in \N_{>0}: a + b < a + c \implies b < c$

and so $+$ is left cancellable on $\N_{>0}$ for $<$.

From Natural Number Addition is Commutative and Right Cancellable Commutative Operation is Left Cancellable:
 * $\forall a, b, c \in \N_{>0}: a + b = a + c \implies b = c$

So $+$ is both right cancellable and left cancellable on $\N_{>0}$.

Hence the result.