Before proving the theorem the book introduces a new definition called implicit equality.
What does exactly the below equality mean? $$ A^{=}x=b^{=} $$ What is inside this set? $$ \{x \in \mathbb{R}^n \colon A^{=}x = b^{=}\} $$ I encountered this notation in Michele Conforti, Integer Programming Book, published by Springer
The book first defines within the n-dimensional space of consideration an included subspace, which is defined by all equalities out of $Ax ≤ b$ (which as subsytem gets denoted by $A^= x = b^=$). This is the space where the polytope, being then defined therein by the inequalities out of $Ax ≤ b$ (which as subsytem gets denoted by $A^< x ≤ b^<$), truely lives.
Consider this example: try to describe the triangle with vertices $(0,0,0)$, $(1,0,0)$, and $(0,1,0)$. I.e. a 2D polygon being described within a 3D space. It is defined by the system $x≥0$, $y≥0$, $x+y≤1$, and $z=0$. Here the first 3 relations are the defining inequalities within the subspace defined by the fourth relation.
--- rk