I know that the product of two Gaussian functions is also Gaussian function. This is stated in Wikipedia, but I might need to cite a classical (text)book/paper stating this result. A book containing the proof is welcome. Could you name such a book?
The product of two normal (Gaussian) random variables is not normal. That would be convenient, but it's not true. Here are a some related theorems that are true:
For more on distribution relationships, see this chart.
If you multiply two normal PDFs you get the joint PDF of a multivariate normal, i.e. if f(x) is a normal PDF then g(x, y) defined by f(x)g(y) is a multivariate normal PDF, but in that case you're multiplying densities and not random variables.
Update: I looked back at your question and I see you said Gaussian "function," rather than "random variable", so perhaps you had in mind the PDF result I mentioned above. If so, just write down the definition of multivariate normal and there it is.
Update 2: Here's an explicit formula for the the product of two normal pdfs being proportional to another normal pdf: blog post.
Correction required in the answer given by Jhon D. Cook.