I understand that a ray parallel to the optical axis will be refracted and go through the focus, and a ray passing the vertex will continue straight ahead. The first statement is a consequence of thin lens formula and lensmaker's equation (or based their derivations). The second statement is a consequence of thin lens.
Now, in the plot, the three red rays originated from the tip of the object on the left hand side of the lens will meet at the tip of the real imagine on the right hand side of the lens. I understand this is consistent and can be shown as a result from the geometric layout of the plot and Newton's version of thin lens formula: $x_1x_2=f^2$.
Considering a thin lens consisting of two spherical surfaces, for paraxial rays, one can derive the lensmaker's equation etc. My question is: how to show that a fourth arbitrary ray (which is still paraxial but is not parallel to the axis neither go through the vertex, not shown in the plot) from the tip of the object should also go pass the tip of the image?
Maybe I have missed something obvious. However, I understand that the derivation of lensmaker's equation in a standard textbook (by explicitly using Snell's law or by using Fermat's principle) only shows how light rays emitted from a fixed point on the optical axis will all meet on another point on the optical axis. Here, the tip of the object involves a small deviation from the optical axis. It is not clearly to me, how to show these light rays are get focused onto the same image point, with higher order (therefore negligible) deviations? Many thanks!