What is the voltage divider rule?
We are looking at an AC amplifier. 

The DC part of the amplifier is only about setting up the Q-Point, or Working Point of the transistor.

Once that is determined, the actual AC workings can be looked at. 

I would propose you get hold of a complete book on basic electronics. The text you showed us here expects quite a bit of knowledge in the reader.

But lets see how this works out: 


Fig. 4.25: We describe the circuit. 
Fig. 4.27: The step showing the DC Part of the input side
Fig. 4.28: What remains if we look at AC -> VCC is replaced with a shortcut. This is a way to calculate the internal resistance of the network we are looking at. This is only a hypothetical step in a mathematical technique. 
"Applying the Voltage Divider Rule", meaning no other thing than "seeing how in 4.28 the Resistances are in parallel, we calculate the parallel resistance using the well known formula. But lets call it a rule". 
This all happens expecting the reader knows how to apply the Thevenin Theorem, which doubtlessly was explained earlier in the book. 
Fig. 4.29 Lets draw the resulting Network, showing the black box we are transforming. 
Eq 4.30 Calculating Ib using the Theremin replacement, this also happens in DC. 
Fig. 4.30 Two steps in this one: First, 4.29 is replaced by the appropriate Thevenin replacement. Secondly, it is applied to the base of the transistor. 
Equation 4.31 has very little to do with the whole path so far, it is simply the summation of the voltages on the output side. 


In fact, Msr. Thevenin was a French pioneer of electronics, read the Wikipedia article. He was not at all mentioned in my studies as an electrical engineer.