# Query in the proof of greedy manipulation theorem (of a voting scheme)

Paper being referred to: http://www.cs.cmu.edu/~arielpro/15896/docs/paper9.pdf (The Computational Difficulty of Manipulating an Election).

I have a query in Theorem 1 of this paper; specifically, in the last argument of its proof. In the proof, the last argument is that $$S(P,c)\geq S(P,u)$$ (by the previous implications) and $$S(P,c) (because placing $$u$$ while completing $$P$$ would make $$c$$ lose). I understand that placing $$u$$ in $$P$$ would make $$c$$ lose; however, I don't understand why that means that $$S(P,c). Can someone clarify this?

My understanding is that the author intends to say that $$u$$ would become the winner, and hence (by the responsiveness property of the score function), $$S(P,x). However, why does $$c$$ not being the winner by putting $$u$$ mean that $$u$$ is the winner? Can't it be possible that there is some other candidate (alternative) $$u'\in U$$ who becomes the winner by placing $$u$$?

For this, what I've thought is that till before placing $$u$$, $$S(P,c)\geq S(P,x)\mbox{ }\forall x\in C\backslash U$$; and we don't know what the final $$S(P,x)$$ will be for any $$x\in U$$. My conjecture then is that placing $$u$$ could only give us information about $$S(P,u)$$ and not about any $$x\in U\backslash \{u\}$$; however, is this conjecture correct? If so, why? Can it not be possible that placing $$u$$ there could give us information about $$S(P,x)$$ for some $$x\in U\backslash\{u\}$$? What property of the score function $$S(P):C\rightarrow \mathbb{R}$$ precludes this (or at least precludes $$S(P,x)$$ from getting definitely larger than $$S(P,c)$$, for any $$x\in U\backslash\{u\}$$)?