Well, every line divides the plane into two, say left and right. Regardless of the number of simultaneous lines, each and every point on the plane must be either left or right of each individual line. If we colour each point according to how many lines it is left of, say red for left of an even number of lines, and blue for left of an odd number of lines, then by induction, every time we cross a line, in any direction, anywhere in the plane, the colour will change, thus satisfying the colouring conditions.