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

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