let x be object ; :: according to VALUED_0:def 9 :: thesis: ( not x in dom (r (#) f) or not (r (#) f) . x is real )
assume x in dom (r (#) f) ; :: thesis: (r (#) f) . x is real
then (r (#) f) . x = r * (f . x) by Def5;
hence (r (#) f) . x is real ; :: thesis: verum