The Kelmans-Seymour conjecture III: 3-vertices in $K_4^{-}$

Let $G$ be a 5-connected nonplanar graph and let $x_1,x_2,y_1,y_2\in V(G)$ be distinct, such that $G[\{x_1,x_2,y_1,y_2\}]\cong K_4^{-}$ and $y_1y_2\notin E(G)$\mathrm{.}Weshowthatoneofthefollowingholds:$G-x_1$contains$K_4^-$,or$G$containsa$K_4^-$inwhich$x_1$isofdegree2,or$G$containsa$TK_5$inwhich$x_1$isnotabranchvertex,or${x_2,y_1,y_2}$ may be chosen so that for any distinct $z_0,z_1\in N(x_1)-\{x_2,y_1,y_2\}$, $G-{x_1v:v\notin {z_0, z_1,x_2, y_1,y_2}}$contains$TK_5$. This result will be used to prove the Kelmans-Seymour conjecture.