The kernel killed by the arrow-radical complementary projection

theorem MagnitudeConjecture.BoundQuiver.StringPresentation.arrowRadicalComplementProjection_ker {k A Q : Type u} [Field k] [Ring A] [Algebra k A] [Fintype Q] [Quiver Q] [(x y : Q) → Fintype (x ⟶ y)] (P : StringPresentation k A Q) {x y : Q} (a : x ⟶ y) : (P.arrowCokernelRadicalLinearMap a).ker ≤ (P.arrowRadicalComplementProjection a).ker