Not necessary, because the legitimacy of the government is irrelevant. What matters is whether most people support Brooks; their support for Brooks isn’t affected by the legitimacy or illegitimacy of the government.

Not necessary, because the overall amount of corruption in the government is irrelevant. What matters is whether most people support Brooks; the level of corruption in the government doesn’t affect whether people support Brooks. Although most supporters perceive Brooks as guilty of corruption, that’s a separate issue from the overall level of corruption in the government.

Not necessary, because the argument isn’t based on the degree of overlap between Brooks’s political positions and McFarlane’s. Even if there is some overlap, it’s still the case that most supporters of McFarlane thinks Brooks is guilty of corruption, and most opponents will oppose Brooks because he agreed to join McFarlane’s government.

Necessary, because if this were not true — if half or more people are NOT supporters and NOT opponents of McFarlane — then there’s a huge portion of the country that the premises do not address. We have no reason to think that those other people would not support Brooks. So we can’t conclude that most people in the country won’t support Brooks.

Not necessary, because whether Brooks is guilty of the charge doesn’t change the fact that most supporters of McFarlane perceive Brooks as guilty of corruption.