Working in a simplicial and constructive context, a new spectral system is defined that relates Serre and Eilenberg-Moore spectral sequences associated to a principal simplicial fibration. The two Eilenberg-Moore spectral sequences (the one where the homology of the fiber is the output, and the other where the homology of the base is computed) are used in our construction. Explicit computer programs are developed, enhancing the Kenzo computer algebra tool to implement that spectral system.