Strict Upper Bound for Composition of Ackermann-Péter Functions

Theorem
For all $x, y, z \in \N$:
 * $\map A {x + y + 2, z} > \map A {x, \map A {y, z}}$

where $A$ is the Ackermann-Péter function.

Proof
We have:
 * $x + y + 1 > y$

giving us:
 * $\paren 1 \quad \map A {x + y + 1, z} > \map A {y, z}$

by Ackermann-Péter Function is Strictly Increasing on First Argument.

Therefore: