The Laws of Ohm (1826) and Kirchhoff's laws (1847) are the basis for the theory of electric circuits. Using them when calculating electrical circuits, we need to create and solve a system of equations for circuit currents and nodal voltages. Kirchhoff's Theorem (1847) on finding currents of branches does not require the compilation and solution of this system of equations. Belov G. A. and Zakharov V. G. (2003) re-proved the Kirchhoff's Theorem (1847) and supplemented it with eight rules for calculating electric circuits. With the help of these rules, the currents of branches and nodal voltages are formed from a loop or nodal determinant and an electric circuit. At the same time, duplicate operations occur in the process of forming expressions of currents and voltages according to these rules. In this paper, we prove the theorems of the Fast Kirchhoff method about a complete and truncated tree, about a unique branch and loop, about common branches and loops, as well as the formulas of the Fast Kirchhoff method for generating currents of branches and nodal voltages.In the Fast Kirchhoff method, when calculating an electric circuit, there are no duplicate operations when generating currents of branches and node voltages, and this reduces the calculation time.