Here we show the proofs alluded to in the section on analyzing , after the balancing transformation. Theorem 4.7.5 of [4, p. 188] asserts that if is a narrowing antifunnel for the differential equation , and if there is a function such that , in U, and such that , then there is a unique solution remaining in U as . Applied to equation (12), we find

for t > 0, so with , the uniqueness criterion is satisfied. This shows that there is a unique solution remaining in each antifunnel.