278 9. TYPE I SINGULARITIES
9.9 is essentially Theorem 24.7 of [63]; the minor extension is that one as-
sumes that the point is Type I c-essential rather than that the singularity
is Type I and the point is c-essential. The alternate proof we gave of the
boundedness of F was suggested to us by Mao-Pei Tsui. The nonnegativity
of the sectional curvatures of ancient solutions was observed by Hamilton.
Theorem 9.19, which does not seem to appear explicitly in print, is analogous
to Corollary B3.5 in [65]. Proposition 9.23 is Theorem 26.1 of [63].