Extension of induction principle proof. i was wandering if is there an extension of the induction principle whether number the integer variables are more than 1?
Example... if i need to prove that $p(n)$ is true then i would start by prove $p(0)$ and assuming for each $k <= n$ is true i'd try to prove $p(n+1)$ is also true...
Is there some generalization whether the statement is of the form $p(n,m)$? and so on and so forth...
 A: As the comments indicate, it is often possible to do induction on each variable separately. Where that isn't useful, you have to define some "successor" relation between $(m, n)$ tuples that allow the induction to go through. One example is the following. Define:
$\begin{align}
   C(m, n)
      = \frac{(2 m)! (2 n)!}{n! m! (m + n)!}
\end{align}$
Prove that $C(m, n)$ is always an integer. 
To do this, we first prove that $C(m, 0)$ is an integer for $m \ge 0$, then prove a relation between $C(m, n), C(m + 1, n), C(m, n + 1)$ that implies that if the first two are integers, so is the third. But:
$\begin{align}
C(m, 0)
  &= \frac{(2 m)! 0!}{0! m! m!} = \binom{2 m}{m}
\end{align}$
which certainly is an integer.
Now do an induction over $n$ to prove $C(m, n)$ is always an integer.
Base: If $n = 0$, we are done by the above.
Induction: We assume $C(m, n)$ is an integer for all $m \ge 0$, and consider:
$\begin{align}
C(m + 1, n) + C(m, n + 1)
  &= \frac{(2 m + 2)! (2 n)!}{(m + 1)! n! (m + n + 1)!}
         + \frac{(2 m)! (2 n + 2)!}{m! (n + 1)! (m + n + 1)!} \\
  &= \frac{(2 m)! (2 n)!}{m! n! (m + n + 1)!}
               \cdot \left(
                       \frac{(2 m + 1) (2 m + 2)}{m + 1}
                         + \frac{(2 n + 1) (2 n + 2)}{n + 1}
                     \right) \\
  &= \frac{(2 m)! (2 n)!}{m! n! (m + n + 1)!}
               \cdot 4 \cdot (m + n + 1) \\
          &= 4 C(m, n)
\end{align}$
As claimed, if $C(m, n)$ and $C(m + 1, n)$ are integers, so is $C(m, n + 1)$. Together with the base, this implies $C(m, n)$ is always an integer.
As a final remark, I haven't been able to find an example of a direct "successor" relation between pairs. This is the closest I've seen.
