Theorem 7.1 — the nonlinear 3-step lift-and-project equals split convexification, $P_j(K) = \mathrm{conv}(K \cap \{x_j \in \{0,1\}\})$

This is Theorem 7.1 of Balas's Disjunctive Programming. Let $K = \{x \in \mathbb R^n : \tilde A x \ge \tilde b\}$ be the LP relaxation of a mixed 0-1 program with 0-1 index set $N'$, the system containing the bound rows $x \ge 0$ and $x_j \le 1$ ($j \in N'$), and let $j \in N'$. Multiply $\tilde A x \ge \tilde b$ by $1 - x_j$ and by $x_j$, linearize $y_i := x_i x_j$ ($i \ne j$) and $x_j^2 := x_j$, obtaining $M_j(K)$, and let $P_j(K)$ be its projection onto $x$. Then

Formalization Note. The retired version allowed an arbitrary system $Ax \ge b$ and an arbitrary coordinate $j$; with no rows, $P_j(K) = \mathbb R^n$ while the hull lies in $0 \le x_j \le 1$. Throughout Chapter 7 the book works with $K = \{x \in \mathbb R^n : Ax \ge b,\ x \ge 0,\ x_j \le 1,\ j \in N'\} = \{x : \tilde A x \ge \tilde b\}$: the bound constraints are rows of the system that every construction multiplies. This is now the explicit hypothesis HasBoundRows A b N' (the system contains the rows $x_k \ge 0$ for every $k$ and $-x_j \ge -1$ for every $j \in N'$); it is satisfiable with nonempty $K$ (e.g. the unit box) and excludes the disproof's instance with no rows. The coordinate $j$ is a 0-1 variable ($j \in N'$), as in the book.