let C be non empty set ; :: thesis: for f being PartFunc of C,COMPLEX holds
( |.f.| " {0 } = f " {0 } & (- f) " {0 } = f " {0 } )
let f be PartFunc of C,COMPLEX ; :: thesis: ( |.f.| " {0 } = f " {0 } & (- f) " {0 } = f " {0 } )
A1: dom |.f.| = dom f by VALUED_1:def 11;

now

let c be Element of C; :: thesis: ( ( c in |.f.| " {0 } implies c in f " {0c } ) & ( c in f " {0c } implies c in |.f.| " {0 } ) )
thus ( c in |.f.| " {0 } implies c in f " {0c } ) :: thesis: ( c in f " {0c } implies c in |.f.| " {0 } )

proof

assume A2: c in |.f.| " {0 } ; :: thesis: c in f " {0c }
then c in dom |.f.| by FUNCT_1:def 13;
then A3: c in dom f by VALUED_1:def 11;
|.f.| . c in {0 } by A2, FUNCT_1:def 13;
then |.f.| . c = 0 by TARSKI:def 1;
then A4: |.(f . c).| = 0 by VALUED_1:18;
c in dom |.f.| by A2, FUNCT_1:def 13;
then f . c = f /. c by A1, PARTFUN1:def 8;
then f /. c = 0c by A4, COMPLEX1:131;
then f /. c in {0c } by TARSKI:def 1;
hence c in f " {0c } by A3, PARTFUN2:44; :: thesis: verum

end;

assume A5: c in f " {0c } ; :: thesis: c in |.f.| " {0 }
then f /. c in {0c } by PARTFUN2:44;
then A6: |.(f /. c).| = 0 by COMPLEX1:130, TARSKI:def 1;
A7: c in dom f by A5, PARTFUN2:44;
then f . c = f /. c by PARTFUN1:def 8;
then |.f.| . c = 0 by A6, VALUED_1:18;
then A8: |.f.| . c in {0 } by TARSKI:def 1;
c in dom |.f.| by A7, VALUED_1:def 11;
hence c in |.f.| " {0 } by A8, FUNCT_1:def 13; :: thesis: verum

end;

hence |.f.| " {0 } = f " {0 } by SUBSET_1:8; :: thesis: (- f) " {0 } = f " {0 }

now

let c be Element of C; :: thesis: ( ( c in (- f) " {0c } implies c in f " {0c } ) & ( c in f " {0c } implies c in (- f) " {0c } ) )
thus ( c in (- f) " {0c } implies c in f " {0c } ) :: thesis: ( c in f " {0c } implies c in (- f) " {0c } )

proof

assume A9: c in (- f) " {0c } ; :: thesis: c in f " {0c }
then A10: c in dom (- f) by PARTFUN2:44;
(- f) /. c in {0c } by A9, PARTFUN2:44;
then (- f) /. c = 0c by TARSKI:def 1;
then - (- (f /. c)) = - 0c by A10, Th9;
then A11: f /. c in {0c } by TARSKI:def 1;
c in dom f by A10, Th9;
hence c in f " {0c } by A11, PARTFUN2:44; :: thesis: verum

end;

assume A12: c in f " {0c } ; :: thesis: c in (- f) " {0c }
then f /. c in {0c } by PARTFUN2:44;
then A13: f /. c = 0c by TARSKI:def 1;
A14: c in dom f by A12, PARTFUN2:44;
then c in dom (- f) by Th9;
then (- f) /. c = - 0c by A13, Th9;
then A15: (- f) /. c in {0c } by TARSKI:def 1;
c in dom (- f) by A14, Th9;
hence c in (- f) " {0c } by A15, PARTFUN2:44; :: thesis: verum

end;

hence (- f) " {0 } = f " {0 } by SUBSET_1:8; :: thesis: verum