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.