The Equivalence of the Identities: p(q.pr) = (pq.r) and pq.rp = p(qr.p)

Moufang loop <L, .> is defined as a loop that satisfies any one of the identities: pq.rp = (p.qr).p, pq.rp = p(qr. p), (pq.r)q = p(q.rq) or p(q.pr)=(pq.p)r. This definition assumes the equivalence of these identities. We had earlier provided the proof of the equivalence of two of these identities: pq.rp = (p.qr).p and pq.rp = p(qr. p) in [5], using a simple algebraic method and manipulation. In this paper, we continue with our style of proof by providing the proof of the equivalence of another two of these identities: p(q.pr)=(pq.p)r and p(q.pr)=pq.rp.