let z be Quaternion; :: thesis: ( z is Real implies z * <j> = [*0,0,(Rea z),0*] )
assume A1: z is Real ; :: thesis: z * <j> = [*0,0,(Rea z),0*]
then reconsider x = z as Real ;
A2: x = Rea z by Lm1;
z * <j> = [*(Rea (z * <j>)),(Im1 (z * <j>)),(Im2 (z * <j>)),(Im3 (z * <j>))*] by QUATERNI:24
.= [*0,(Im1 (z * <j>)),(Im2 (z * <j>)),(Im3 (z * <j>))*] by A1, QUATERNI:34
.= [*0,0,(Im2 (z * <j>)),(Im3 (z * <j>))*] by A1, QUATERNI:34
.= [*0,0,x,(Im3 (z * <j>))*] by QUATERNI:34
.= [*0,0,(Rea z),0*] by A2, QUATERNI:34 ;
hence z * <j> = [*0,0,(Rea z),0*] ; :: thesis: verum