let z be Quaternion; :: thesis: |.(z *').| = |.z.|
A1: z *' = [*(Rea z),(- (Im1 z)),(- (Im2 z)),(- (Im3 z))*] by Th36;
then A2: Im1 (z *') = - (Im1 z) by Th16;
A3: Im2 (z *') = - (Im2 z) by A1, Th16;
Im3 (z *') = - (Im3 z) by A1, Th16;
hence |.(z *').| = |.z.| by A1, A2, A3, Th16; :: thesis: verum