If there are two lines that coincide with each other, then these two lines will have the same number of different intersection points with the rest of the lines. Assume that there are no lines that coincide with each other and all lines have different number of different intersection points with the rest of the lines, so each of the $2000$ lines has its number of different intersection points with the rest of the lines ranging from $0$ to $1999$ and since every number is different and there are $2000$ lines with $2000$ possibility, we must have one with $0$ intersection and one with $1999$ intersections which is a contradiction. Thus, there must be two lines that have the same number of different intersection points with the rest of the lines.