Which network theorem states "the current through, or voltage across, an element in a linear bilateral network is equal to the algebraic sum of the currents or voltages produced independently by each other"

Superposition  theorem

Explanation:

The superposition theorem states that "the net electrical force on a specific charge is equal to the sum of the vector components of the charges applying forces on it".

This theorem is applicable for both linear and bilateral networks and comes as a consequence of the Coulomb's Law, which describes the force between two punctual electrical charges at rest.

However, Coulomb's law does not tell us anything about what happens if we have more than two electrical charges or if they are not punctual.  For example, if we have three charges aligned and we want to find the force on one of the ends and how the presence of the central electrical charge influences. The evidence from the experiments has shown that this central charge does not affect the force between them, and this can be expressed by the superposition principle which, in other words, is:  

Given a system of punctual charges, the electric force on each of them is the vector sum of the forces due to each of the other charges, as if the rest of the charges were not present.  

That is, we can calculate each term by Coulomb's law ignoring completely the existence of other charges.