Finally we get to proving theorem 9. We reuse the notation and the definitions from the previous two subsections. We aim to show that the Reeb precosheaf \(\mathcal{C}\) defines a \(1\)-homomorphism from \(\mathbf{D}\) to \(\mathbf{C}\). Now in order for this statement even to make any sense, the image of \(\mathbf{D}\) under \(\mathcal{C}\) should lie in \(\mathbf{C}\).
Lemma. The image of \(\mathbf{D}\) under \(\mathcal{C}\) is part of the subcategory \(\mathbf{C}\).
Now let \(f \colon X \rightarrow {\mathbb{R}}\) be a bounded constructible \({\mathbb{R}}\)-space. To proof the previous lemma it suffices to show that \(\mathcal{C} (r + \mathcal{E} f)\) and \(\mathcal{C} (r + \mathcal{R} \mathcal{E} f)\) are an object of \(\mathbf{C}\) for all \(r \in (-\infty, \infty]\).
We consider the case \(r = 0\) first. By lemma 19 the precosheaf \(\mathcal{C} \mathcal{E} f\) is an object of \(\mathbf{C}\). By lemma 30 from the last appendix the projection \(\mathcal{E} f\) from \({\operatorname{epi}}f\) to \(\overline{{\mathbb{R}}}\) is a constructible \(\overline{{\mathbb{R}}}\)-space. This in conjunction with lemma 6 yields the following
Lemma. The homomorphism \((\mathcal{C} \circ \pi \circ \mathcal{E})_f\) from \(\mathcal{C} \mathcal{E} f\) to \(\mathcal{C} \mathcal{R} \mathcal{E} f\) is an isomorphism of precosheaves.
Now \(\mathbf{C}\) is closed under isomorphisms and thus also \(\mathcal{C} \mathcal{R} \mathcal{E} f\) lies in \(\mathbf{C}\).
We continue with the case of \(r\) not necessarily being \(0\). To this end let \(g \colon Y \rightarrow (-\infty, \infty]\) be a continuous function, possibly an object of \(\mathbf{D}\). We note that for any \(r \in (-\infty, \infty]\) we have \(r + g = \mathcal{S}((\infty, -r)) (g)\), by definition of \(\mathcal{S}\). We now recall that we also defined the endofunctors associated to the smoothing functor \(\mathcal{S}'\) for \(\mathbf{D}\) on the whole category of set-valued precosheaves on \(\overline{D}\) and not just \(\mathbf{D}\). With \(\mathbf{D}\) being invariant under these endofunctors, lemma 23 follows from the following
Lemma. For all \(\mathbf{a} \in -D\) we have \((\mathcal{C} \circ \mathcal{S}(\mathbf{a}))_g = (\mathcal{S}'(\mathbf{a}) \circ \mathcal{C})_g\).
Proof. For \((a, b) \in -D\) we have \(\Delta \circ \mathcal{S}((a, b))(g) = S^{(-b, -b)} \circ \Delta \circ g\) and this implies the claim.
Next we show that \(\mathcal{C}\) is compatible with the natural transformations of the smoothing functors. To this end let \(\mathbf{a}, \mathbf{b} \in -D\) with \(\mathbf{a} \preceq \mathbf{b}\).
Lemma. We have \((\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_f = (\pi^2_* \circ \mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{E})_f\).
Proof. First we set \((a', a) := \mathbf{a}\) and \((b', b) := \mathbf{b}\). Now let \(r \in {\mathbb{R}}\). If we unravel the definitions we obtain \[ (\overline{\mathcal{S}}(\mathbf{a}) \pi^2_* \mathcal{C} \mathcal{E} f)( [-\infty, r) ) = \Lambda({\operatorname{epi}}f \cap X \times [-\infty, r + a)) \] and \[ (\overline{\mathcal{S}}(\mathbf{b}) \pi^2_* \mathcal{C} \mathcal{E} f)( [-\infty, r) ) = \Lambda({\operatorname{epi}}f \cap X \times [-\infty, r + b)) . \] Let \(i\) be the inclusion of \({\operatorname{epi}}f \cap X \times [-\infty, r + a)\) into \({\operatorname{epi}}f \cap X \times [-\infty, r + b)\) and let \[ \tau \colon {\operatorname{epi}}f \cap X \times [-\infty, r + a) \rightarrow {\operatorname{epi}}f \cap X \times [-\infty, r + b), (p, t) \mapsto (p, t - a + b) , \] then \[ (\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_{f ~ [-\infty, r)} = \Lambda(i) \] and \[ (\pi^2_* \circ \mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{E})_{f ~ [-\infty, r)} = \Lambda(\tau) . \] Now let \((p, t) \in {\operatorname{epi}}f \cap X \times [-\infty, r + a)\), then \(\{p\} \times [t, t - a + b]\) is contained in \({\operatorname{epi}}f \cap X \times [-\infty, r + b)\). Moreover we have \((p, t), (p, t - a + b) \in \{p\} \times [t, t - a + b]\), hence \(\Lambda(i) = \Lambda(\tau)\).
Corollary. We have \((\mathcal{S}'(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{C} \circ \mathcal{E})_f = (\mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{E})_f\).
Proof. This follows from \(\pi^2_*\) being a \(1\)-homomorphism and being full and faithful on \(\mathbf{C}\).
Corollary. We have \((\mathcal{S}'(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{C} \circ \mathcal{R} \circ \mathcal{E})_f = (\mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{R} \circ \mathcal{E})_f\).
Proof. By the lemmata 22 and 25 and the previous corollary we have the commutative diagram \[ \xymatrix@R+=4pc@C+=6pc{ \mathcal{S}'(\mathbf{a}) \mathcal{C} \mathcal{E} f \ar[r]^{ (\mathcal{S}'(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{C} \circ \mathcal{E})_f } \ar[d]|-{ (\mathcal{S}'(\mathbf{a}) \circ \mathcal{C} \circ \pi \circ \mathcal{E})_f } & \mathcal{S}'(\mathbf{b}) \mathcal{C} \mathcal{E} f \ar[d]|-{ (\mathcal{S}'(\mathbf{b}) \circ \mathcal{C} \circ \pi \circ \mathcal{E})_f } \\ \mathcal{S}'(\mathbf{a}) \mathcal{C} \mathcal{R} \mathcal{E} f \ar[r]_{ (\mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{R} \circ \mathcal{E})_f } & \mathcal{S}'(\mathbf{b}) \mathcal{C} \mathcal{R} \mathcal{E} f . } \] Now by lemma 24 the vertical arrows are isomorphisms and thus the naturality of \(\mathcal{S}'(\mathbf{a} \preceq \mathbf{b})\) implies that \((\mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{R} \circ \mathcal{E})_f = (\mathcal{S}'(\mathbf{a} \preceq \mathbf{b}) \circ \mathcal{C} \circ \mathcal{R} \circ \mathcal{E})_f\).
Eventually the previous two corollaries and lemma 25 imply the
Proposition. The functor \(\mathcal{C}\) is a \(1\)-homomorphism from \(\mathbf{D}\) to \(\mathbf{C}\).
Now let \(g \colon Y \rightarrow {\mathbb{R}}\) be another bounded constructible \({\mathbb{R}}\)-space. Then we have the following
Corollary. The interleavings of \(\mathcal{R} \mathcal{E} f\) and \(\mathcal{R} \mathcal{E} g\) with respect to \(\mathcal{S}\) are in bijection with the interleavings of \(\mathcal{C} \mathcal{E} f\) and \(\mathcal{C} \mathcal{E} g\) with respect to \(\mathcal{S}'\).
Proof. We observe that all join trees of bounded constructible \({\mathbb{R}}\)-spaces lie in a \(-D\)-subcategory of \(\mathbf{D}\). By the previous proposition \(\mathcal{C}\) yields a \(1\)-homomorphism from this \(-D\)-category to \(\mathbf{C}\). Moreover corollary 8 implies that \(\mathcal{C}\) is full and faithful on this category and thus the interleavings of \(\mathcal{R} \mathcal{E} f\) and \(\mathcal{R} \mathcal{E} g\) are in bijection with those of \(\mathcal{C} \mathcal{R} \mathcal{E} f\) and \(\mathcal{C} \mathcal{R} \mathcal{E} g\). Now by lemma 24 these are isomorphic to \(\mathcal{C} \mathcal{E} f\) and \(\mathcal{C} \mathcal{E} g\) and this implies the claim.
Now we can finally proof theorem 9. To this end let \(f \colon X \rightarrow {\mathbb{R}}\) and \(g \colon Y \rightarrow {\mathbb{R}}\) be continuous maps with \(X\) and \(Y\) smooth and compact manifolds.
Proof (Theorem 9). Let \(\delta > 0\), by Bröcker and Jänich (1973 Satz 14.8) there are smooth functions \(f' \colon X \rightarrow {\mathbb{R}}\) and \(g' \colon Y \rightarrow {\mathbb{R}}\) with \({\lVert f - f' \rVert}_{\infty} \leq \frac{\delta}{8} \geq {\lVert g - g' \rVert}_{\infty}\). Further there are Morse functions \(f'' \colon X \rightarrow {\mathbb{R}}\) and \(g'' \colon Y \rightarrow {\mathbb{R}}\) with \({\lVert f' - f'' \rVert}_{\infty} \leq \frac{\delta}{8} \geq {\lVert g' - g'' \rVert}_{\infty}\) by Milnor (1963 corollary 6.8). These two results together yield \({\lVert f - f'' \rVert}_{\infty} \leq \frac{\delta}{4} \geq {\lVert g - g'' \rVert}_{\infty}\). By corollary 14, corollary 4, proposition 16, proposition 21, and the previous corollary, we have \[ \begin{split} \mu(\mathcal{C} \mathcal{E} f, \mathcal{C} \mathcal{E} g) & \leq \mu(\mathcal{C} \mathcal{E} f, \mathcal{C} \mathcal{E} f'') + \mu(\mathcal{C} \mathcal{E} f'', \mathcal{C} \mathcal{E} g'') + \mu(\mathcal{C} \mathcal{E} g'', \mathcal{C} \mathcal{E} g) \\ & \leq {\lVert f - f'' \rVert}_{\infty} + \mu_J (\mathcal{R} \mathcal{E} f'', \mathcal{R} \mathcal{E} g'') + {\lVert g'' - g \rVert}_{\infty} \\ & \leq \mu_J (\mathcal{R} \mathcal{E} f'', \mathcal{R} \mathcal{E} f) + \mu_J (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) + \mu_J (\mathcal{R} \mathcal{E} g, \mathcal{R} \mathcal{E} g'') + \frac{\delta}{2} \\ & \leq {\lVert f'' - f \rVert}_{\infty} + \mu_J (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) + {\lVert g - g'' \rVert}_{\infty} + \frac{\delta}{2} \\ & \leq \mu_J (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) + \delta . \end{split} \] Similarly we have \(\mu_J (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) \leq \mu(\mathcal{C} \mathcal{E} f, \mathcal{C} \mathcal{E} g) + \delta\). Since \(\delta > 0\) was arbitrary, we have \(\mu_J (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) = \mu(\mathcal{C} \mathcal{E} f, \mathcal{C} \mathcal{E} g)\).
The proof for \(M\) and \(M_J\) is completely analogous.