Ackermann-Péter Function at (2,y)

Theorem
For every $y \in \N$:
 * $\map A {2, y} = 2 y + 3$

where $A$ is the Ackermann-Péter function.

Proof
Proceed by induction on $y$.

Basis for the Induction
Suppose $y = 0$.

Then:

Induction Hypothesis
Suppose:
 * $\map A {2, y} = 2 y + 3$

We want to show that:
 * $\map A {2, y + 1} = 2 \paren {y + 1} + 3 = 2 y + 5$

Induction Step
By the Principle of Mathematical Induction, the result holds for all $y \in \N$.