"Uniqueness" is, by its nature, a "negative"- there does not exist another object with the same properties. And, typically, negative statements are most simply proved by "contradiction".
To prove (6), that the point bisecting a line is unique (of course, "uniqueness" here refers to the uniqueness of the object, not the method of construction), assume there are two points, p1 and p2, that both bisect the line segment. Assume that p1 is, of the two, closer to one endpoint of the segment. That means, of course, that p2 is closer to the other endpoint. Now use the fact that these are both "bisectors" to show a contradiction.