The book value is Total Assets minus Total Liabilities and so if you increase the Total Assets without changing the Total Liabilities the difference gets bigger and thus higher.
Consider if a company had total assets of $4 and total liabilities of $3 so the book value is $1. Now, if the company adds $2 to the assets, then the difference would be 4+2-3=6-3=3 and last time I checked 3 is greater than 1.
On definitions, here are a couple of links to clarify that side of things.
Equity = Assets - Liabilities
From Ready Ratios:
Shareholders Equity = Total Assets – Total Liabilities
Shareholders Equity = Share Capital + Retained Earnings – Treasury
Depending on what the reinvestment bought, there could be several possible outcomes. If the company bought assets that appreciated in value then that would increase the equity. If the company used that money to increase sales by expanding the marketing department then the future calculations could be a bit trickier and depend on what assumptions one wants to make really.
If you need an example of the latter, imagine playing a game where I get to make up the rules and change them at will. Do you think you'd win at some point? It would depend on how I want the game to go and thus isn't something that you could definitively say one way or the other.