set R = f ^ ;
assume not f ^ is non-empty ; :: thesis: contradiction
then 0 in rng (f ^ ) by RELAT_1:def 9;
then consider x being set such that
A1: x in dom (f ^ ) and
A2: (f ^ ) . x = 0 by FUNCT_1:def 5;
dom (f ^ ) = dom f by Th2;
then reconsider a = f . x as non zero real number by A1;
not a " is empty ;
hence contradiction by A1, A2, RFUNCT_1:def 8; :: thesis: verum