The homotopy-wedge theorem for near-prisms

Let $X$ be a near-prism in direction $i$, and define $B_i(X)=\operatorname{del}_i X\setminus\operatorname{link}_i X$. Suppose that $\operatorname{del}_i X$ is homotopy equivalent to a wedge of spheres

$$\operatorname{del}_i X\simeq\bigvee_j S_j^{r_j}.$$

Near-prism homotopy theorem. Then $X$ is homotopy equivalent to a wedge of spheres, specifically

$$X\simeq\bigvee_j S_j^{r_j+c_{i,j}},$$

where $c_{i,j}$ is the number of $j$-dimensional cells in $B_i(X)$. This gives a recursive description of the homotopy type of near-prisms from that of their deletion and the cells in the associated boundary difference.