let x be object ; :: according to VALUED_0:def 8 :: thesis: ( not x in dom (f1 + f2) or (f1 + f2) . x is ext-real )
assume x in dom (f1 + f2) ; :: thesis: (f1 + f2) . x is ext-real
then (f1 + f2) . x = (f1 . x) + (f2 . x) by Def888;
hence (f1 + f2) . x is ext-real ; :: thesis: verum