Prove that

Since four force and four momentum are perpendicular their vector dot product equals zero.

So the three dimensional parts of force and momentum are related as follows:

Multiply both sides by dt

Three dimensional displacement equals the product of the three velocity vector and the differential time scaler dt.

Now the scalar differential element of work (dw) will equal the dot product of the two three dimensional vectors of force and displacement.

So the differential scalar of work (dw) is proportional to the differential scalar of mass (dm). Here the square of the speed of light (c) is the constant of proportionality.

So integrate the work done in moving from an initial work and mass to a final work (E) and mass (m).

Thus