let z be quaternion number ; :: thesis: ( z is Real implies z * <i> = [*0 ,(Rea z),0 ,0 *] )
assume A1: z is Real ; :: thesis: z * <i> = [*0 ,(Rea z),0 ,0 *]
then reconsider x = z as Real ;
A2: x = Rea z by Lm2;
z * <i> = [*(Rea (z * <i> )),(Im1 (z * <i> )),(Im2 (z * <i> )),(Im3 (z * <i> ))*] by QUATERNI:24
.= [*0 ,(Im1 (z * <i> )),(Im2 (z * <i> )),(Im3 (z * <i> ))*] by A1, QUATERNI:33
.= [*0 ,x,(Im2 (z * <i> )),(Im3 (z * <i> ))*] by QUATERNI:33
.= [*0 ,x,0 ,(Im3 (z * <i> ))*] by QUATERNI:33
.= [*0 ,(Rea z),0 ,0 *] by A2, QUATERNI:33 ;
hence z * <i> = [*0 ,(Rea z),0 ,0 *] ; :: thesis: verum