R^3/Cross product/Oriented orthonormal basis/Fact/Proof

Proof

Let

and

Due to fact  (2), we have

Due to fact  (3), we have

and, because of fact  (1), we have

According to fact  (6), is perpendicular to and to ; therefore,

with some , as this orthogonality condition defines a line. Because of fact  (5) and the condition, we get

hence,

Using fact  (3), we obtain and . Altogether we get

and this is the claim.