Following a combinatorics answer that uses identities e.g. Vandermonde Identity, $\binom{-n}{k}=(-1)^k\binom{n+k-1}{k}$, ... My question may be really elementary, simple, or obvious, and yet the solution misses me.
My question is on a combinatorics answer to the question Proving $\sum\limits_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$. 
I follow most of the mathematics in the answer, such as the Chu-Vandermonde identity, identity $(\mathbf{1})$ below,

$$\binom{-n}{k}=(-1)^k\binom{n+k-1}{k}\tag{1}$$

and all the steps except for two, namely, how does $(\mathbf{6})$ follow from $(\mathbf{5})$?

$$ \begin{align} &=(-1)^{M-n}\binom{-k-n-2}{M-n}\tag{5}\\
&=\binom{M+k+1}{M-n}\tag{6}\\ \end{align}$$

The following shows my attempt to answer that question.
I noticed that $(\mathbf{5})$ takes the same form as the $(-1)^k\binom{n+k-1}{k}$ of identity $(\mathbf{1})$. For example, both have $(-1)$ raised to a power $k$, and multiplied by a binomial coefficient that is indexed by $k$. Now, if the expression $$(-1)^{M-n}\binom{-k-n-2}{M-n}\tag{5}$$
is somehow in a form compatible with identity $(\mathbf{1})$, then we can substitute $(\mathbf{5})$ into the RHS of $(\mathbf{1})$, and the LHS result of $(\mathbf{1})$  will hopefully equal $(\mathbf{6})$.
Still, it's not immediately obvious to me if or how $(\mathbf{5})$ can be transformed into the RHS of $(\mathbf{1})$, so that we can apply the identity $(\mathbf{1})$ to show $(\mathbf{5}) = (\mathbf{6})  $ .
What algebraic manipulations, renaming, substitutions, identities or otherwise, are needed to show $(\mathbf{5}) = (\mathbf{6})$?   
 A: Notice how identity (1) takes the form:
$$\binom{-n}{k}=(-1)^k\binom{n+k-1}{k} = (-1)^k\binom{k-(-n)-1}{k}$$
Compare with
$$(-1)^{M-n}\binom{-k-n-2}{M-n}=(-1)^{M-n}\binom{(M-n)-(M+k+1)-1}{M-n}$$
It follows from (1) that 
$$(-1)^{M-n}\binom{(M-n)-(M+k+1)-1}{M-n} = \binom{M+k+1}{M-n}$$
and hence
$$(-1)^{M-n}\binom{-k-n-2}{M-n}=\binom{M+k+1}{M-n}$$
A: I hadn't noticed until today how straightforward it is to see how $\mathbf{(6,5)}$ are an instance of $(\mathbf{1})$. If you notice that the RHS of  $$\binom{-n}{k}=(-1)^k\binom{n+k-1}{k}\tag{1}$$ hasn't many requirements, and allows for many possibilities  to satisfy this form. The RHS allows any $k \in \mathbb{Z}^+$, has $k$ appear as the exponent to $(-1)$, and observe that $\binom{n+k-1}{k}$ only asks for some $n \in \mathbb{Z}$, and this imposes no relationship on $n, k$. So there isn't much special about $n \ \text{or}\ k$.
It does not take much to form an instance of $(\mathbb{1})$, so without much thought we can show that $(\mathbb{5})$ is an instance of the RHS of $(\mathbb{1})$ like so

Let $-n = M + k + 1, k = M - n$.  Then  $n + k - 1 = -(M+k+1)+(M-n)-1 = -k - n - 2$.

and without anything but variable assignments and substitution we have shown $(\mathbf 5, 6)$ are an instance of $(\mathbf{1})$. One might notice that the concluding statement $\binom{n+k+1}{k} = \binom{-k-n-2}{M - n}$ looks like a contradiction when we compare the binomial coefficients' indices, but keep in mind the $n$ defined in $(\mathbb{1})$ is not the same $n$ as in $(\mathbb{5,6})$, which is defined in the context of a larger $7$-step answer. 
