Binary 2s Compliment Applied Twice Gives Original - How?

At the moment the maths behind binary 2s compliment seems like voodoo to me.

If I take 0011 (+3) and "flip and add 1", I get 1101 (-3)

If I apply the same process to 1101, ("flip and add 1"), I get back to the original positive value.

How is this so?

If we look at the numbers in an unsigned way, flipping a binary number $x$ on $n+1$ bits is computing $(2^{n+1} -1) - x = M - x$.

Proof for that: $x = \sum_0^n b_i2^i$. $x$ flipped : $\sum_0^n (1-b_i)2^i = \sum_0^n 2^i -\sum_0^n b_i2^i = (2^{n+1} - 1) - x$

Therefore, flip and add one : $y = (M - x) + 1$

Again : $z = (M - y) +1 = (M - ((M - x) + 1)) + 1$

$z = (M - (M - x) - 1) + 1$

$z = M - M + x - 1 + 1$

$z = x$

StackExchange Computer Science Q#102383, answer score: 3