let z be quaternion number ; :: thesis: 2 * (Rea z) = Rea (z + (z *' ))
A1: z = [*(Rea z),(Im1 z),(Im2 z),(Im3 z)*] by QUATERNI:24;
a: z *' = [*(Rea z),(- (Im1 z)),(- (Im2 z)),(- (Im3 z))*] by QUATERNI:43;
z + (z *' ) = [*((Rea z) + (Rea z)),((Im1 z) + (- (Im1 z))),((Im2 z) + (- (Im2 z))),((Im3 z) + (- (Im3 z)))*] by A1, a, QUATERNI:def 7
.= [*(2 * (Rea z)),0 ,0 ,0 *] ;
hence 2 * (Rea z) = Rea (z + (z *' )) by QUATERNI:23; :: thesis: verum