Problem:

Ronald Fisher discovered what he, with humility, called the Fundamental Theorem of Natural Selection. This theorem says (in its modern terminology):

The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genetic variance in fitness at that time.

As I understand it, it sounds alike the standard equation that we learn in the first class of Introduction to evolutionary biology

R = S \cdot \frac{V_G}{V_p}

In words: the response to selection equals the selection differential times the genetic variance of the trait under consideration divided by the total phenotypic variance of the trait under consideration

Required:

Question : How can we prove/demonstrate that Fisher's fundamental theorem of natural selection holds true?

I don't ask for empirical evidences that support this claim but for a theoretical/mathematical proof/demonstration of this claim.

Show the pedigree and your calculation.