forward implication of the first statement.

Perform induction on the degrees of $u$,$v$ and $w$. The induction base is straightforward. Now consider the induction step. By our assumption, $\beta(u) > \beta(v)$ and hence $\beta(S(u)) > \beta(S(v))$. Therefore by induction we obtain, $\beta(u \centerdot S(w)) > \beta(v \centerdot S(w))$ and $\beta(S(u) \centerdot w) > \beta(S(v) \centerdot w)$. Summing up and using Theorem \[t:derivation\] and Lemma \[l:reduction\], we get $\beta(u \centerdot w) > \beta(v \centerdot w).$

The backward implications of both the statements are again similar. To see the backward implication of the first statement, assume the contrary, that is, either $\beta(u) < \beta(v)$ or $\beta(u) = \beta(v)$. Then the forward implications, which we just proved, give a contradiction.

\[l:switch\] Let $u$ and $v$ be [**cd**]{}-monomials. Then we have, $\beta(u \centerdot v) = \beta(u \centerdot v^*) = \beta(u^* \centerdot v)$ and $\beta(u \centerdot v) = \beta(v \centerdot u)$.

We first prove the first statement by an induction on the size of the [**cd**]{}-monomial. It is enough to show only the first equality. The second one follows by the symmetry in our argument. The induction base is provided by the statements $\beta({e}\centerdot v) = \beta({e}\centerdot v^*)$ and $\beta(u \centerdot {e}) = \beta(u \centerdot 