  We give a constructive proof that coordinate transformations exist which
raise the regularity of the gravitational metric tensor from $C^{0,1}$ to
$C^{1,1}$ in a neighborhood of points of shock wave collision in General
Relativity. The proof applies to collisions between shock waves coming from
different characteristic families, in spherically symmetric spacetimes. Our
result here implies that spacetime is locally inertial and corrects an error in
our earlier RSPA-publication, which led us to the false conclusion that such
coordinate transformations, which smooth the metric to $C^{1,1}$, cannot exist.
Thus, our result implies that regularity singularities, (a type of mild
singularity introduced in our RSPA-paper), do not exist at points of
interacting shock waves from different families in spherically symmetric
spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case
of such shock wave interactions but our proof strategy differs fundamentally
from that used by Israel and is an extension of the strategy outlined in our
original RSPA-publication. Whether regularity singularities exist in more
complicated shock wave solutions of the Einstein Euler equations remains open.
