With the APS boundary condition, the appearance in (In.6) of the eta invariant is a little less striking, since it represents just a part of the in­ formation in the projection Q+. The condition (In.8) reflects the squareintegrability of an extension of the solution into t < 0, i.e. to the whole of the original manifold X. It will not be encountered below1. As already noted, the invertibility of the operator 3+, which in general for an exact 6-metric is not quite as simple as (In.7), is attacked directly and its gener­ alized inverse and the associated heat kernels are shown to be elements of the appropriate space of 6-pseudodifferential operators. 4. Preliminaries to the proof.