Let us suppose that the transformation equations from a rectangular (x, y, z) system to the and systems are given by

and

Then there will exist a transformation directly from the system to the system defined by  

and conversely.

The radius vector to point P is

in the system and

in the system.

From 4) we get

and from 5)

Then

From 3) above we obtain

Substituting 7) into 6) and equating coefficients of on both sides we obtain

The vector A is given by

and

where C₁, C₂, C₃ and are the contravariant components of A in the two systems.

Substituting 8) into 10) we get

Equating the coefficients of in 9) and 11) we get

End of proof.