You have only suggested that there must be only one. You have not proven it.
1) Draw a line segment with a midpoint.
2) Note that the two sides have identical length
3) Put another midpoint on the line segment. I doesn't matter where.
4) Note that the distance between the two midpoints measures other than zero.
5) Call the three segments so defined A, B, and C. B is between the two midpoints.
6) If both are midpoints, then mA = mB + mC AND mA + mB = mC
7) What does this say about the length of B?
8) What does that say about where the second "midpoint" really is?