$x = XOR(a_0, a_1, ...\space a_n), \exists \space j \in [[0, n]] a_j \space xor (a_j - x) \space xor\space x = 0$, prove me right or wrong

Question

I tell you that given any family of natural number:

$$a_0,\space a_1,\space ...\space a_n$$

(of any finite lenght), posing

$$x = XOR(a_0,\space a_1,\space ...\space a_n)$$

We have $$\exists\space j \in [[0, n]]\space a_j\space xor\space (a_j - x)\space xor\space x = 0$$

Prove me right or wrong.

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Now, I expect you might want to express the natural numbers as n-element vectors of $\mathbb F_2$ where $\mathbb F_2 = \lbrace 0, 1\rbrace$, with + as the xor operator over $\mathbb F_2$ as well as over $\mathbb F_2^n$, so be free to use this (these) notation(s) without defining it (them).

Answer 1

Apparently, I was wrong.

The family $(1, 2, 5, 10)$, with $x = 12$ is a counter-example. None of the numbers $b$ respect $b \space xor \space (b - x) \space xor \space x = 0$.