let z be quaternion number ; :: thesis: 2 * (Rea z) = Rea (z + (z *' ))
A1: ( z = [*(Rea z),(Im1 z),(Im2 z),(Im3 z)*] & z *' = [*(Rea z),(- (Im1 z)),(- (Im2 z)),(- (Im3 z))*] ) by QUATERNI:24, QUATERNI:43;
A2: z + (z *' ) = [*((Rea z) + (Rea z)),((Im1 z) + (- (Im1 z))),((Im2 z) + (- (Im2 z))),((Im3 z) + (- (Im3 z)))*] by A1, QUATERNI:def 7
.= [*(2 * (Rea z)),0 ,0 ,0 *] ;
thus 2 * (Rea z) = Rea (z + (z *' )) by A2, QUATERNI:23; :: thesis: verum