let z, z1 be quaternion number ; :: thesis: ( z is Real implies z * z1 = z1 * z )
assume A1: z is Real ; :: thesis: z * z1 = z1 * z
then reconsider x = z as Real ;
( Rea z = x & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 ) by A1, Lm0;
hence z * z1 = z1 * z ; :: thesis: verum