Ackermann-Péter Function at (1,y)

Theorem
For every $y \in \N$:
 * $\map A {1, y} = y + 2$

where $A$ is the Ackermann-Péter function.

Proof
Proceed by induction on $y$.

Basis for the Induction
Suppose $y = 0$.

Then:

This is the basis for the induction

Induction Hypothesis
This is the induction hypothesis:

Suppose that:
 * $\map A {1, y} = y + 2$

We need to show that:
 * $\map A {1, y + 1} = y + 3$

Induction Step
This is the induction step:

Thus, by Principle of Mathematical Induction, the result holds for all $y$.