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FLP86's famous proof regarding impossibility of async distributed processes with a single fault assumes in the proof of the third lemma the existence of an event $e'$ such that the neighbor configurations $C_0$ and $C_1$ can be related as

$C_1 = e'(C_0)$.

I don't get how this is possible, as it seems to me like $e'$ carries out a state transition from a 0-valent configuration to a 1-valent configuration. In addition to this being counter intuitive the proof of case 1 of lemma 3 clearly states that any successor of a 0-valent configuration has to be a 0-valent configuration. What am I missing here?

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At least in this part of the article, the character C is used for bivalent configurations. The index $0$ for the bivalent $C_0$ was chosen simply because $e(C_0)$ is 0-valent and the index $1$ in $C_1$ was chosen because $e(C_1)$ is 1-valent, even though $C_0$ and $C_1$ are bivalent neighbors (when applying $e'$).

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