What is the proof that we can use the location of virtual image in the lens formula to get the location of the image (convex/concave lens)? The following problem might make my question more clear. The proof for the case of plane mirror is quite easy using ray diagrams but I'm getting contradictory results when I try to use the same for convex/concave lens.
In this image "p" is the location of virtual object and "r" is the location of the image formed after reflection . I would like to restate my question as to Why can we substitute the distance of "P" in the lens formula to get the location of the image formed after refraction by the arriving rays? What is the proof?