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You still don't get it, Em, do you? Brown put himself forward for selection as leader of the Labour party and NO ONE ELSE did. That is, he was unopposed. In the British system in general, not just in the Labour Party, guess what that means...it means he wins! You'll probably find it's exactly the same at your local Con-club, WRI or whatever when committee-members etc are being chosen.
In horse-racing, it's called a "walkover", which is now used generally in English for any situation where there is a simple victory. You must have heard of it.
Macmillan, Douglas-Home and Major, as I've already pointed out, all took over the Prime Ministership in precisely the same circumstances and were, therefore, equally 'unelected'.
I did spend some years in education, but - really, for you - here endeth the lesson!