User:Etheryte/Sandbox

Theorem of Induction of Two Variables
If $\{P(i,j)\}$ is a set of statements such that
 * 1) $P(0,j)$ is true for $j \geq 0$ and $P(i,0)$ is true for $i \geq 0$ and
 * 2) $(P(i-1,j) \wedge P(i,j-1)) \rightarrow P(i,j)$ for $i \geq 1$, $j \geq 1$,

then $P(m,n)$ is true for all nonnegative integers $m$ and $n$.


 * : $\S 3.5$: Theorem $3.67$

Proof
Still needed.

Theorem
$10$ is the only triangular number which is the sum of two consecutive odd squares:


 * $10 = 1^2 + 3^2$