set R = f ^ ;
assume not f ^ is non-empty ; :: thesis: contradiction
then 0 in rng (f ^) ;
then consider x being object such that
A1: x in dom (f ^) and
A2: (f ^) . x = 0 by FUNCT_1:def 3;
dom (f ^) = dom f by Th2;
then reconsider a = f . x as non zero Real by A1, ORDINAL1:def 16;
not a " is zero ;
hence contradiction by A1, A2, RFUNCT_1:def 2; :: thesis: verum