identity at an arbitrary [**cd**]{}-monomial and then use induction on its degree to show that they yield the same value.

The first step is to check directly that the identity holds at the [**cd**]{}-monomials ${e}, 1, {{\bf c}}$ and ${{\bf d}}$. This is straightforward to check using Figure \[f:val\]. To complete the induction step, we begin by evaluating the RHS at $u v$ and expanding using the inductive definitions of ${\hat \Delta}$ and $\hat G$. The induction step is shown below. $$\begin{array}{r c l}
RHS_{u v} & = & ({\operatorname{id}}{\otimes}{\hat G} + {\hat G} {\otimes}{\operatorname{id}})
({\hat \Delta}(u v)) \\
                & = & ({\operatorname{id}}{\otimes}{\hat G} + {\hat G} {\otimes}{\operatorname{id}})
({\hat \Delta}(u) v + u {\hat \Delta}v).
\end{array}$$ On expanding further, we obtain four terms, two of which we write down explicitly. $$\begin{array}{r c l}
({\operatorname{id}}{\otimes}{\hat G})({\hat \Delta}(u) v) 
    & = & ({\operatorname{id}}{\otimes}{\hat G})({\hat \Delta}(u)) v +
{\hat \Delta}(u) \hat G(v) -
{\hat \Delta}(u) {{\bf c}}v -
u {\otimes}v,\\
({\hat G} {\otimes}{\operatorname{id}})({\hat \Delta}(u) v)
    & = & ({\hat G} {\otimes}{\operatorname{id}})({\hat \Delta}(u)) v.
\end{array}$$ The correction term $- u {\otimes}v$ in the first expression accounts for the difference between the terms $({\operatorname{id}}{\otimes}{\hat G})(u {\otimes}{e}v)$ and $({\operatorname{id}}{\otimes}{\hat G})(u {\otimes}{e}) v.$

The remaining two terms 