  Let $P$ and $Q$ be two orthogonal projections on a separable Hilbert space,
$\calH$. Wang, Du and Dou proved that there exists a unitary, $U$, with
$UPU^{-1} =Q, \quad UQU^{-1} = P$ if and only if $\dim(\ker P \cap \ker(1-Q)) =
\dim(\ker Q \cap \ker(1-P))$ (both may be infinite). We provide a new proof
using the supersymmetric machinery of Avron, Seiler and Simon.
