5.3.3. Theorem 5.2.

d( , A i +t A j ,, A j , )=d( , A i ,, A j , )

where ij  and t is a scalar.

Proof

The case t=0  is trivial so that we shall assume t0 .  By axiom 1, we have

        d( , A i ,,t A j , )=td( , A i ,, A j , )

By axiom 2, we have

        d( , A i ,,t A j , )=d( , A i +t A j ,,t A j , )  

=td( , A i +t A j ,, A j , )              [axiom 1 used]

Comparing with the 1st equation finishes the proof.