Here are three things that may or may not be different:
    A. Montana's biggest city (represents appetites, the biggest soul-part)
    B. Montana's capital (represents rational soul-part, supposed to be in charge)
    C. Montana's most famous city (for spirited soul-part, seeks fame and recognition)

How to find whether these are one, two, or three cities?  It could be, for example, that the capital, the biggest city, and the most famous are all the same city!  And it might be that, in fact, you really don't know whether they are or aren't the same.

Suppose we manage to show that Montana's biggest city (appetites) is different from its capital (reason).  Then there are definitely two cities; but are there three?  It still might be that the most famous city (spirited part) is the same as the biggest city (appetites), so that we really have only two cities.  And it might also be that the most famous city (spirited part) is the capital (reason).  We've shown A and B are different; but C and A might be the same.  And also C and B might be the same.

So now suppose that we manage to show that the most famous city (spirited part) is not the biggest city (appetites).  We still haven't proved that there are three cities, because it might be that the most famous city (spirited part) is the capital (reason).  Now we've shown that A and B are different, and that C and A are different. But C and B still might be the same.

Finally, suppose we manage to show that the most famous city and the capital are different: B and C are different.  Now, and only now, have we managed to prove that these really are three different cities.  A isn't B, A isn't C, and B isn't C.  Appetites aren't reason, appetites aren't spirit, and spirit
isn't reason.

Ah, but how do we show that two city-names really apply to different cities instead of to the same city?  Well, what if we found that they disagreed with each other, or even were at war with each other?  Wouldn't that show they were different?  Seems it would!

Or consider balls moving in a box - if all we know is that a ball named A and a ball named B are moving in the same direction (suppose we can't actually see them), then A and B could, for all we know, be different designations or names for the same ball.  But if we know that A and B are moving in different directions at the same time, then they have to be different; there have to be two of them.

Socrates doesn't think soul-parts have to conflict with each other.  But they can, and often do.  That's enough to show that they are different.  Like different cities, or nations.  They can be cooperative and harmonious, but often they aren't.