Natural Number Addition is Cancellable

Theorem
Let $\N_{> 0}$ be the $1$-based natural numbers.

Let $+$ be addition on $\N_{>0}$.

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 $\N_{>0}$.

Proof
By Ordering on $1$-Based Natural Numbers is Trichotomy, one and only one of the following holds:
 * $a = b$
 * $a < b$
 * $b < a$

Suppose $a < b$.

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

By Ordering on $1$-Based 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 $1$-Based 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 $\N_{>0}$.

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.