Hey Mcoolta.

The best way to prove these kinds of results is to look at the moment generating functions:

Moment-generating function - Wikipedia, the free encyclopedia

After looking at the following:

Log-normal distribution - Wikipedia, the free encyclopedia

It appears that you need to use the characteristic function, however the same deal applies: if the product of the two functions has the same form then you can use this to show that the addition of two distributions has the same distribution form.