Let  be the velocity field of an incompressible irrotational fluid and  be that of any other incompressible fluid motion with same normal component velocity  at the boundary of the domain, where  is the unit vector of the bounding surface (and, if the domain extends to infinity,  there). Then the difference between the kinetic energy is given by
can be rearranged to give 
Since  is irrotational and the domain is simply-connected, a single-valued velocity potential exists, i.e., . Using this, the second integral in the above equation can be written as
The second integral is identically zero for steady incompressible fluid, i.e., . Applying the Gauss theorem for the first integral we find
where the surface integral is zero since normal component of velocities are equal there. Thus, one concludes
or in other words, , where the equality holds only if , thereby proving the theorem.