Obviously, a5+a4 and a5-a4 cannot be expressed linearly by a 1, a2 and a3.

If it can be expressed, there are k 1, k2, k3 such that a5+a4 = k1a1+k2a2+k3a3.

Because of the existence of m 1, m2 and m3, a4 = m1a1+m2a2+m3a3,

Therefore, a5 = (k1-m1) a1+(k2-m2) a2+(k3-m3) a3 can be obtained.

This contradicts the known fact that a5 cannot be expressed linearly by a 1, a2, a3.

Therefore, a5+a4 cannot be expressed linearly by a 1, a2, a3. Similarly, it can be proved that a5-a4 cannot be expressed.