Assume you want to prove an identity such as
$$\sum_{k=m+1}^{n}A(k,m)-B(k,m)=S(m)+T(n,m)\qquad\text{for } n,m\in
\mathbb{Z},n,m\geq 0.$$
Added: I applied mathematical induction on $m,n$ to prove it. I am unsure because up to now I have seen it applied to properties depending on a single variable only.
Question: does application of two inductive arguments, one on $m$ and the
other on $n$, guarantee the validity of such a proof?
Here are some induction principles for two variables:
$\forall x,y. P(x,y) \Rightarrow P(x,y+1)$
$\forall x,y. P(x,y)$
and
$\forall x,y. P(x+1,y) \Rightarrow P(x,y+1)$
$\forall x,y. P(x,y)$
Suppose you are trying to prove a family of statements $P(x, y)$. This is the same as proving the family of statements $F(x)$, where $F(x) = \forall y : P(x, y)$. Each statement $F(x)$ can be proven by induction on $y$ (for fixed $x$), and then you can prove $P(x, y)$ by induction on $x$. You might want to try proving
$${n+1 \choose k+1} = {n \choose k+1} + {n \choose k}$$
this way.
But actually you can be much trickier than this. Sometimes it suffices to induct on $x + y$, for example.