let r be real number ; :: according to PARTFUN3:def 4 :: thesis: ( r in rng (f ^ ) implies 0 <= r )
set R = f ^ ;
assume r in rng (f ^ ) ; :: thesis: 0 <= r
then consider x being set such that
A1: x in dom (f ^ ) and
A2: (f ^ ) . x = r by FUNCT_1:def 5;
dom (f ^ ) = (dom f) \ (f " {0 }) by RFUNCT_1:def 8;
then x in dom f by A1, XBOOLE_0:def 5;
then f . x in rng f by FUNCT_1:def 5;
then reconsider a = f . x as real non negative number by Def4;
not a " is negative ;
hence 0 <= r by A1, A2, RFUNCT_1:def 8; :: thesis: verum