let x be object ; :: according to VALUED_0:def 9 :: thesis: ( not x in dom (f ^) or (f ^) . x is real )
set F = f ^ ;
assume x in dom (f ^) ; :: thesis: (f ^) . x is real
then (f ^) . x = (f . x) " by Def2;
hence (f ^) . x is real ; :: thesis: verum