This is what we have so far, because we are trying to prove that (0, infinity) is a subset of D^P and D^p is a subset of (0, infinity).

Part 1:

If a exists (0, infinity)

Then according to the Definition:

a + a = 2a, which must be positive because 2 times a positive number is a positive number.

Additionally, a * a = a^2, which is obviously positive because any number squared is positive.

Part 2:

We proved in Part 1 that a exists in (0, infinity). Now, we are letting a belong to D^p.

**Pay attention that this is not necessarily the same a!**
Thus, by the last part of the definition, we see that if a exists D^p, then exactly one of the following is true:

a=0, a belongs to D^p, or -a belongs to D^p.

Since we proved that a belongs to D^p. This means that only a can belong to D^p.

**You assumed(!) that , not proved! You don't know anything about this a!**
Which means that a is not 0 or negative.