HOL theorem-proving system Reviews

There are no examples of anything. 1. A complete proof of a common theorem of mathematics. 2. A new theorem being created rather than using existing ones. 3. Translating a proof into normal mathematical notation. (This would show that it really works.) This should be built in. It would make the system infinitely better. How about the well-known proof that there are irrational X and Y such that X^Y (X to the power of Y) is rational? How about proving the square root of 2 is irrational? Is there any truth to HOL or is it just Big Speculation? Charlie Volkstorf