It seems to be asserting that, for all possible k, if the first k input values x_1, ... x_k are replaced by the mean x, then the mean doesn't change.
Sorry what?
(1 2 3 4 5 6) -> arithmetic mean is x = 3.5.
Ok, let k = 3; replace first 3 elements by 3.5:
(3.5 3.5 3.5 4 5 6) -> arithmetic mean is 4.25
Unless I'm totally daft, there has to be a typo there somewhere?
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Edit, ah, ok. Returning to this, I see where my eyes fooled me. The term on the left goes to x_k. Thus x is a "sub-mean" representative value which is of the first k values. If the first k values are replaced by their "sub-mean", then the "master mean" is undisturbed.
That is very intuitive.
In the case of an arithmetic mean, it's basically a distribution situation: a certain total quantity is distributed into n slots. If we transfer an amount from any slot to any other, the mean does not change. We can choose a subset of the slots, and redistribute their subtotal among them that they have an equal value. That value will be their respective mean. Since we do not add anything to the total, the mean doesn't change.
What isn't so obvious is that this generalizes to the other kinds of means, like geometric.