For uniformly accelerating linear motion with zero initial velocity, the difference of displacement between adjacent and equal time intervals is constant? Why

For uniformly accelerating linear motion with zero initial velocity, the difference of displacement between adjacent and equal time intervals is constant? Why

If you make a V-T image, no matter the initial velocity is zero or not, then the displacement within the same time interval should be some trapezoids with larger area, and the area of each trapezoid is the displacement. The difference between two adjacent trapezoids must be equal, so let's prove it