Prove of Elliptic Curve Addition Operation

This section describes how to prove that the addition operation on an elliptic curve can be successfully performed geometrically.

To prove that each and every addition operation on an elliptic curve can be successfully performed, we need to prove the following:

1. Every straight line that passes through two points P and Q on an elliptic curve must intersect with the same curve at a third point -R. Otherwise we will not be able to find -R. I need to do more research to prove this.

2. Every straight line that passes through two points P and Q on an elliptic curve must intersect with the same curve in at most three points. Otherwise we may find multiple possible points of -R. This can be approved based on the first property of elliptic curves described earlier.

3. Every point -R on an elliptic curve must have an x-axis symmetrical point R on the same curve. Otherwise we will not be able to find R. This can be approved based on the second property of elliptic curves described earlier.