Introduce a generalization of the Pappus theorem and the Pascal theorem

Question

The theorem proved in A chain of six circles associated with a conic

Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a hyperbola Let $1'$ be arbitrary point in the hyperbola. The circle $(121')$ meets the conic at point $2'$. The circle $(232)$ meets the hyperbola again at $3'$, define points $4', 5', 6'$ similarly. Let circle $(121')$ meets the circle $(454')$ at $A, B$, Let circle $(232')$ meets the circle $(565')$ at $C, D$. Let circle $(343')$ meets the circle $(616') $ at $E, F$. Then six points $A, B, C, D, E, F$ lie on a circle.

Special case:

1. If $1'$ at $\infty$ the theorem is Pascal theorem.

2. If the hyperbola is two lines, and $1'$ at $\infty$ the theorem is Pappus theorem