Proving Newton's Law of Universal Gravitation

[abcd8989]

Given F is directly proportional to the product of m1 and m2. F is also inversely proportional to the r^2. F, m1, m2 and r are real numbers.
Why we can link the above two variations together and say that F=km1m2/r^2, where k is the proportionality constant? Aren't the the variations independent? How to prove that variations can be linked like that?
(This question is essentially Newton's Law of Universial Gravitation.)

 

[Ted Sand]

When you write F = (k1)(m1)(m2) where 'k1' is a constant, you are assuming that you keep everything else constant and only may vary the masses.
What if I define k1 = G/r^2, where G is a constant and r is a constant?

If you write F = (k2)/r^2 where 'k2' is a constant, you are assuming you may vary the radius whilst keeping everything else constant.
What if I define k2 = G(m1)(m2), where G is a constant and m1, m2 are constant?

These two equations are both consistent with F = G(m1)(m2)/r^2