Multiplying monomial polynomials

Let $\mu+\nu$ be a partition $(\la_1+\mu_1,\la_2+\mu_2,\dots,\la_n+\mu_n)$.

Lemma 1  

$$m_\la m_\mu = m_{\la+\mu}+\sum_{\nu<\la+\mu}a^\nu_{\la,\mu} m_\nu,\qquad a^\nu_{\la,\mu}\in{\Bbb Z}_{\geq 0}.$$

Exercise:Proof the lemma. $\;\Box$