$(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}\ $ [Clement's theorem] The Wiki page on Twin Primes says

The pair $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod
{m(m+2)}$.

This is obviously connected to Wilson's Theorem. Can anybody provide a proof for that, along the lines of thought given here,
Wilson's Theorem/Proofs/Prime modulus – another proof?
 A: By Wilson's Theorem, if $m$ is prime, then $(m-1)!+1\equiv 0\pmod{m}$, and therefore 
$$4[(m-1)!+1]+m \equiv 0\pmod{m}.\tag{$1$}$$
If $m+2$ is prime, then $(m+1)!+1\equiv 0\pmod{m+2}$, again by Wilson's Theorem. But since $m+1\equiv -1\pmod{m+2}$, and $m\equiv -2\pmod{m+2}$, we have 
$(m+1)!\equiv (-1)(-2)(m-1)!\equiv 2(m-1)!\pmod{m+2}$, and therefore
$$4[(m-1)!+1]+m \equiv 2(m+1)!+2+m+2\equiv 0\pmod{m+2}.\tag{$2$}$$
Since $m$ is odd, the numbers $m$ and $m+2$ are relatively prime, and therefore by $(1)$ and $(2)$ we have $4[(m-1)!+1]+m\equiv 0\pmod{m(m+2)}$.
The proof of the converse follows similar lines, this time using the fact that the Wilson condition is sufficient for primality.
A: Follows immediately (mechanically!) by W = Wilson's theorem and (Easy) CRT as below,
using $\!\bmod m\!+\!2\!:\ \color{#0a0}{(m\!+\!1)!} = \smash{\underbrace{(m\!+\!1)m}_{\large \ \ (-1)(-2)}(m\!-\!1)!\equiv \color{#0a0}{2(m\!-\!1)!}}\,\ $ [Wilson reflection]
$\begin{align}
m\!+\!2\ {\rm prime} &\smash{\overset{\small \rm W\!}\iff} \overbrace{\color{#0a0}{(m\!+\!1)!}}^{\large\color{#0a0}{2(m-1)!}}\equiv -1\!\!\!\pmod{\!m\!+\!2}\smash{\overset{\color{#0a0}{\times\ 2}}\iff} 4(m\!-\!1)!\equiv -2\!\!\!\pmod{\!m\!+\!2}\\[.5em]
m\ {\rm prime} &\smash{\overset{\small \rm W\!}\iff} (m\!-\!1)!\equiv -1\!\!\!\pmod{\!m}\ \ \ \smash{\overset{\times\ 4}\iff}\ \ \ 4(m\!-\!1)!\equiv -4\!\!\!\pmod{\!m}\\
\end{align}$
$$\overset{\small \rm CRT}\iff 4(m\!-\!1)!\equiv -4 + m\underbrace{\left[\dfrac{2}{\color{#c00}m}\bmod m\!+\!2\right]}_{\large \color{#c00}m\ \equiv\ -2}\equiv\,\bbox[5px,border:1px solid #c00]{-4\!-\!m}\,\ \pmod{\!m(m\!+\!2)}$$
The method  generalizes to prime $k$-tuples.
