let z be quaternion number ; :: thesis: ( z is real implies Rea z = z )
assume z is real ; :: thesis: Rea z = z
then z in REAL by XREAL_0:def 1;
then ( Rea z = z & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 ) by Lm17;
hence Rea z = z ; :: thesis: verum