convex polytopes – Number of vertices in a polyhedron

Consider polytopes

$$A_1[x_{1,1},dots,x_{1,m_1},z_{1}]’leq b_1$$
$$A_2[x_{2,1},dots,x_{2,m_2},z_{2}]’leq b_2$$
$$B[z_{1},z_{2},z]’leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.

We eliminate $z_{1}$ to $z_{2}$ by Fourier Motzkin and get a new polyhedron
$$C[x_{1,1},dots,x_{2,m_2},z]’leqtilde c.$$

Is the number of vertices in the new polyhedron $leq v_1v_2$ if $m_1=m_2$?