Can someone clarify me this proof for this theorem?
Theorem: If are tri different points of line and are points of line such that , then there exists unique point such that and . Also, point belongs to line and ordering of points on line matches ordering of points on line .
I will show proof for order of points .
Proof: If and are points of ray such that , and then it follows so because of there is point that meets conditions of theorem.
I don't understand why we need point at all. If we asume that there is point such that , and then it follows which is the same proof.