Cauchy's integral theoremCauchy's integral theorem and Cauchy's integral formulaCauchy's integral formula.
It's really an example of a jewellery-type and tool-type theorem at the same time. It can be introduced and proved for students that even don't know about functions of complex variables in 20 minutes. And other 10 minutes can be spend to say how many applications and generalizations these results have in theory of functions and applied mathematics.