  Let $A \in Z^{m \times n}$, $rank(A) = n$, $b \in Z^m$, and $P$ be an
$n$-dimensional polyhedron, induced by the system $A x \leq b$.
  It is a known fact that if $F$ is a $k$-face of $P$, then there exist at
least $n-k$ linearly independent inequalities of the system $A x \leq b$ that
become equalities on $F$. In other words, there exists a set of indices $J$,
such that $|J| \geq n-k$, $rank(A_{J}) = n-k$, and
  $$
  A_{J} x - b_{J} = 0,\quad \text{for any $x \in F$}.
  $$
  We show that a similar fact holds for the integer polyhedron
  $$
  P_{I} = conv.hull\bigl(P \cap Z^n\bigr),
  $$
  if we additionally suppose that $P$ is $\Delta$-modular, for some $\Delta \in
\{1,2,\dots\}$. More precisely, if $F$ is a $k$-face of $P_{I}$, then there
exists a set of indices $J$, such that $|J| \geq n-k$, $rank(A_{J}) = n-k$, and
  $$
  A_{J} x - b_{J} \overset{\Delta}{=} 0,\quad \text{for any $x \in F \cap
Z^n$},
  $$ where $x \overset{\Delta}{=} y$ means that $\|x - y\|_{\infty} < \Delta$.
In other words, there exist at least $n-k$ linearly independent inequalities of
the system $A x \leq b$ that almost become equalities on $F \cap Z^n$. When we
say almost, we mean that the slacks are not greater than $\Delta-1$. Using this
fact, we prove the inequality
  $$
  |vert(P_I)| \leq 2 \cdot \binom{m}{n} \cdot \Delta^{n-1},
  $$ for the number of vertices of $P_I$, which is better, than the state of
the art bound for $\Delta = O(n^2)$.
