\pi_1 (a) \pi_1 (b) = \pi (a) |_{K_1} \pi (b) |_{K_1} = \pi (a) \pi (b) |_{K_1} = \pi (ab) |_{K_1} = \pi_1 (ab) .
So (π1, V, K1) is also a Stinespring representation of Φ and has the additional property that K1 is the closed linear span of π(A) V*H.