It is just a definition, but it's the only definition of the complex exponential function which is well behaved and is equal to the real variable function on the real line.
Also, every identity about analytical functions on the real line also holds for the respective complex function (excluding things that require ordering). They should have probably explained it.
Denazification was a years long process of removing nazis from positions of cultural and political power and reversing the impacts of nazi propaganda on the German population (done by the occupying Allies). The Nuremberg trials by themselves were not enough for Germany to become a normal country again.