I need a proof (using a computer is allowed) of the following identity, which involves only binomial sums and products and quotients.
For $1\leq m\leq k \leq n$: $$A_{k,n,m} + \binom{k}{m}= B_{k,n,m}$$
Where
$$A_{k,n,m} = \sum_{i=1}^m \sum_{j=i}^{k-1} \frac{k-j}{k-i} \binom{k-2i}{m-i}\binom{k-i}{i}\binom{j-1}{i-1}\binom{n-k+j-1}{j}$$
and
$$B_{k,n,m} = \sum_{h=0}^{n-k-1} \frac{\binom{n}{k-m}\binom{n}{m}}{\binom{n}{k}}\binom{m-1+h}{h}\binom{k-m+h}{h}\frac{m(h+k-n+1)+n-k}{(h+1)(n-k)}$$
I am aware of the fact that algorithms such as Wilf-Zeilberger's are useful to do such jobs, but I am not being able to access to any computer version of them due to software incompatibility, so a help would be really appreciated (i.e. providing the computational proof and or certificate is perfectly fine).