Cw homotopy equivalence inclusions are strong deformation retracts

First prove the relative inverse assertion: if $f:(X,A)\to (Y,A)$ is a homotopy equivalence, equals the identity on $A$, and both inclusions have HEP, let $g$ be an inverse and $h_t:gf\simeq {\mathrm{i}\mathrm{d}}_X$. Extend $h_t{\mathrm{\mid }}_A$ over $Y$ by HEP, starting at $g$, to obtain $g_t$ with $g_1{\mathrm{\mid }}_A=\mathrm{i}\mathrm{d}$. CW inclusions have the needed HEP by F1.

Concatenate the homotopy $g_{1-2t}f$ for $0\le t\le 1\mathrm{/}2$ with $h_{2t-1}$ for $1\mathrm{/}2\le t\le 1$. It runs from $g_{1}f$ to the identity, and on $A$ is a path followed by its reverse. Such a loop contracts relative to its endpoints: if its first-half path is $\gamma$, replace $\gamma (\min (2t,2-2t))$ by $\gamma ((1-u)\min (2t,2-2t))$. HEP for $(X\times I,A\times I)$ extends this homotopy of homotopies. Following the left, top, and right sides of the parameter square now gives $g_{1}f\simeq {\mathrm{i}\mathrm{d}}_X$ relative to $A$. This product HEP follows by taking the product of the HEP retraction $X\times I\to X\times \{0\}\cup A\times I$ with the other interval.

Repeat the preceding adjustment with $g_1$ and $f$ interchanged, obtaining $f_1$ fixed on $A$ with $f_1{g}_1\simeq {\mathrm{i}\mathrm{d}}_Y$ relative to $A$. Then $f_1\simeq f_1{g}_{1}f\simeq f$ relative to $A$, so $f{g}_1\simeq {\mathrm{i}\mathrm{d}}_Y$ relative to $A$. Apply this result with the map $A\hookrightarrow X$: its relative inverse is a retraction, and the relative inverse homotopy is precisely a strong deformation retraction. If $A$ is empty, the existence of an inverse forces $X$ empty.

For a homotopy $U:B\times I\to Z$ from $u_0$ to $u_1$, use the common space $W=Z{\cup }_U(Y\times I)$. The CW prism retraction of $Y\times I$ onto $Y\times \{0\}\cup B\times I$ fixes $B\times I$ and descends to a strong deformation retraction of $W$ onto the first adjunction space, fixed on $Z$. The reversed prism gives the second retraction. These prism retractions can be built cellwise using radial projection of $D^r\times I$ onto its bottom and sides; concatenate over dimensions with the CW weak topology, as in the HEP construction. Composing the two inclusions and retractions gives inverse homotopy equivalences relative to $Z$.