Pascal theorem for three dimensions

Question

A year ago I found the Pascal theorem for three dimentions as follows:

Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ be six arbitrary points lie on $(C_1)$; $B_1$ be arbitrary point on $(C_2)$. Plane $A_iB_iA_{i+1}$ meets $(C_2)$ again at $B_{i+1}$ for $i=1, 2, \cdots, 6$. Then $B_7=B_1$. Let $P_i$ is the plane through $A_i, A_{i+1}, B_i, B_{i+1}$ for $i=1, 2, \cdots, 6$. Then $P_1$ meets $P_4$, $P_2$ meets $P_5$, $P_3$ meets $P_6$ at three lines on the same plane.

My question: How can prove this result and can generalization to higher dimensions?

See also:

• A chain of six circles associated with a conic

• Can generalization of a generalization Pascal theorem, Pappus theorem to Higher Dimensions?