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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

In Figure we redraw Figure using and .
If , choose **C** with ; the unique solution
lying in the antifunnel defined by and
must be either above or beneath it, because
solutions cannot intersect. If it is above, is also above,
contradicting the definition of ; if it is beneath, then so is
, contradiciting the definition of . Thus, every solution
is of the form for some **C**, and there is a unique such
solution for each **C**.

**Figure 13:** Fences and
associated with a solution .

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