Let \(H\) be a complex Hilbert space with inner product \(\langle,\rangle\). Let \(P\) be an orthogonal projection. An operator \(A\) on \(H\) is said to be \(P\)-invertible if and only if there exists an operator \(B\) on \(H\) with the property: \(PAPB=BPAP=P\). The authors of this work prove a multiplicity theorem which replaces a variety of rules used in the theory of the intermediate problem of the first type for eigenvalues of semi- bounded self-adjoint operators on a complex Hilbert space.