set F = f ^ ;
let x be set ; :: according to VALUED_0:def 9 :: thesis: ( not x in dom (f ^ ) or (f ^ ) . x is real )
assume x in dom (f ^ ) ; :: thesis: (f ^ ) . x is real
then (f ^ ) . x = (f . x) " by Def8;
hence (f ^ ) . x is real ; :: thesis: verum