The Argument at Republic 348b-350c

I wish to sharpen and restate a question I posed in the comments as regards the argument in Rep. I, 348b-350c, and then raise another.

The first question is this. One might have thought that the argument can be captured, more or less, as follows:

1. A just person wishes and thinks it right that he have more than an unjust person, but does not wish and think it right that he have more than a just person.
2. An unjust person wishes and thinks it right that he have more than a just person, and also wishes and thinks it right that he have more than an unjust person.
3. Thus, a just person aims to exceed, not his like, but only those unlike him.
4. But an unjust person aims to exceed both his like and unlike.
5. Each of these is to be identified with that class to which he may be likened.
6. A knowledgeable person wishes and claims that he should do more (in a certain domain) than an ignorant person, but does not wish and claim that he should do more than a knowledgeable person.
7. An ignorant person wishes and claims that he should do more (in a certain domain) than a knowledgeable person, and also wishes and claims that he should do more than an ignorant person.
8. Thus, a knowledgeable person aims to exceed, not his like, but only those unlike him.
9. But an ignorant person aims to exceed both his like and unlike.
10. Thus, a just person may be likened to a knowledgeable person; an unjust person to an ignorant person.
11. Thus, a just person is knowledgeable; an unjust person is ignorant.
12. But virtue implies wisdom and knowledge.
13. Thus, injustice is not virtue (as Thrasymachus had claimed).

But there is no mention here of a just or unjust person's having more 'than the just action'. So why does Plato add this extra detail? What work does it do in the argument, as he understands it, and in which way is the above construction faulty or incomplete?

As I said in the comments, it might look as though the argument works just as well (or badly) without the very awkward additional remarks about actions. So then why does Plato go out of his way to include these?

A second question.

Why does Plato twice use the double construction "wishes to have more and thinks it right that he have more" rather than one of these alone? He seems deliberately to use both, but what is the purpose of this? (Yes, I do believe that such details are deliberate.)

dikai/ou mh\ a)cioi= ple/on e)/xein mhde\ bou/letai o( di/kaioj (349c1)

dokei= a)\n ou)=n ti/j soi, w)= a)/riste, mousiko\j a)nh\r a(rmotto/menoj lu/ran e)qe/lein mousikou= a)ndro\j e)n th=| e)pita/sei kai\ a)ne/sei tw=n xordw=n pleonektei=n h)\ a)ciou=n ple/on e)/xein; (349e10-13)