Electrical – Proving the formula for parallel resistors

circuit analysisohms-lawresistanceresistors

I am currently studying The Art of Electronics, third edition, by Horowitz and Hill. Exercise 1.3 asks to prove the formulas for series and parallel resistors. I am given the following relevant information earlier in the chapter:

1. The sum of the currents into a point in a circuit equals the sum of the currents out (conservation of charge). This is sometimes called Kirchhoff’s current law (KCL). Engineers like to refer to such a point as a node. It follows that, for a series circuit (a bunch of two-terminal things all connected end-to-end), the current is the same everywhere.
2. Things hooked in parallel (Figure 1.1) have the same voltage across them. Restated, the sum of the “voltage drops” from A to B via one path through a circuit equals the sum by any other route, and is simply the voltage between A and B. Another way to say it is that the sum of the voltage drops around any closed circuit is zero. This is Kirchhoff’s voltage law (KVL).

I am trying to use this information (and ohm's law $$\ V = IR \$$, of course) to prove the parallel case $$\R = \dfrac{R_1 R_2}{R_1 + R_2} \$$. However, I do not understand how the provided information is sufficient to deduce this.

I am told that this is a good textbook, so perhaps it is just that I am new to electronics, and so I am not understanding something. Or, perhaps it is the case that the authors have failed to provide enough information to solve this problem without external knowledge. I would greatly appreciate it if people would please take the time to explain how it is possible to deduce this with the given information.