A conjecture on binomial factors Can any one help me prove the following conjecture:
\begin{equation}
\sum_{p=1}^{\min(n,m+1)}C_{m+1}^p C_{n-1}^{p-1}=\sum_{p=1}^{\min(m+1,n+1)}C_n^{p-1}C_m^{p-1}=C_{m+n}^n
\end{equation}
Here $C_{m+n}^n=\frac{(m+n)!}{m!\cdot n!}$ is the binomial factor, and $\min(a,b)$ means the smaller one between $a$ and $b$. 
I have verified these two equations for some small values of $n$ and $m$, but it would be great to have a proof. Thank you a lot!

Consider for example
$$
\binom{n+m}{n} = \sum_{p=0}^\infty \binom{n}{p} \binom{m}{p} = \sum_{p=0}^\infty \binom{n}{n-p} \binom{m}{p} = \sum_{p+q=n} \binom{n}{q} \binom{m}{p}.
$$
Partition the set $[n+m]$ into two parts: $A$ of length $n$ and $B$ of length $m$. The left-hand side counts the number of subsets of $[n+m]$ of size $n$. The right-hand side counts the same number, breaking each subset into a part in $A$ (of some size $q$) and a part in $B$ (of size $n-q$).
