Rewrite (hyper.1) as a first order system
The essential ingredient here is the existence of a positive symmetrizer, H > 0 ,
so that multiplication by H on the left gives
Multiplying by we are led to
and the real part of both sides are in fact perfect derivatives, for by the symmetry of H,
and similarly, by the symmetry of , we have
Hence, by integration over the -period we end up with energy conservation, asserting
We note that the positivity of H was not used in the proof and is assumed just for the sake of making (u,Hu) an admissible convex ``energy norm.''