let z be Quaternion; ( z is Real implies z * <j> = [*0,0,(Rea z),0*] )
assume A1:
z is Real
; 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*]
; verum