  In this paper we will consider two curl-curl equation in two dimensions. One
curl-curl problem for a scalar quantity $F$ and one problem for a vector field
$\bf{E}$. For Dirichlet boundary conditions $\bf{n} \times \bf{E} =$ $
\hat{E}_{\dashv}$ on $\bf{E}$ and Neumann boundary conditions $\bf{n} \times
\mbox{curl}$ $F=\hat{E}_{\dashv}$, we expect the solutions to satisfy
$\bf{E}=\mbox{curl}$ $F$. When we use algebraic dual polynomial
representations, these identities continue to hold at the discrete level.
Equivalence will be proved and illustrated with a computational example.
