Derivation of an acceleration

Two boxes are side by side on a floor with friction, from the left a Force acts on them. To be more concise about the picture it is this one on this website:

http://hyperphysics.phy-astr.gsu.edu/hbase/f2m2.html#c1

I dont understand how the acceleration is derived. Especially i dont get which vectors are weighed out against each other. And also from their pictures it is not clear where

I have already tried to "reverse engineer" the given expression for a (on that website):

Step-1: [itex]a = \frac{F- \mu*(m_{1}+m_{2})*g}{(m_{1}+m_{2})}[/itex]

Step-2: [itex]a*(m_{1}+m_{2}) = F- \mu*(m_{1}+m_{2})*g[/itex]

My question here is wether the left side stands for "[itex]F_{net}[/itex]" (or "[itex]F_{result}[/itex]") ? That is could i write instead of Step-2, this one:

Step-3: [itex]F_{net} = F- \mu*(m_{1}+m_{2})*g[/itex]

And the next question is could i replace F with [itex]F_{push}[/itex]?

Step-4: [itex]F_{net} = F_{push}- \mu*(m_{1}+m_{2})*g[/itex]

and the mu-Stuff with

Step-5: [itex]F_{net} = F_{push}- f_{m1}-f_{m2}[/itex]

If so, why didnt they write it on their website, because i think that it is more intuitive than just spitting out the formula for a? Or is my derivation totally wrong?