let z be quaternion number ; :: thesis: ( z is Real implies ( z = Rea z & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 ) )
assume z is Real ; :: thesis: ( z = Rea z & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 )
then reconsider x = z as Real ;
B2: x = [*x,0 *] by ARYTM_0:def 7;
[*x,0 *] = [*x,0 ,0 ,0 *] by QUATERNI:91;
hence ( z = Rea z & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 ) by B2, QUATERNI:23; :: thesis: verum