The superposition theorem, a particular case of the superposition principle, states that in a linear circuit with several voltage and current sources, the current and voltage for any element of the circuit is the algebraic sum of the currents and voltages produced by each source acting independently. The superposition theorem is not applicable to power, because it is a non-linear quantity. Therefore, the total power dissipated in a resistor must be calculated using the total current through (or the total voltage across) it. The theorem proposed and proved in this paper states that in a linear DC network consisting of resistors and independent voltage and current sources, the total power dissipated in the resistors of the network is the sum of the power supplied simultaneously by the voltage sources with the current sources replaced by open circuit, and the power supplied simultaneously by the current sources when the voltage sources are replaced by short-circuit. This means that the power is superimposed. The theorem can be used to simplify the power analysis of DC networks. The analysis results are validated via numerical examples.