let f be complex-valued Function; :: thesis: ( (abs f) " {0 } = f " {0 } & (- f) " {0 } = f " {0 } )

now

let c be set ; :: thesis: ( ( c in (abs f) " {0 } implies c in f " {0 } ) & ( c in f " {0 } implies c in (abs f) " {0 } ) )
thus ( c in (abs f) " {0 } implies c in f " {0 } ) :: thesis: ( c in f " {0 } implies c in (abs f) " {0 } )

proof

assume A1: c in (abs f) " {0 } ; :: thesis: c in f " {0 }
then (abs f) . c in {0 } by FUNCT_1:def 13;
then (abs f) . c = 0 by TARSKI:def 1;
then abs (f . c) = 0 by VALUED_1:18;
then f . c = 0 by COMPLEX1:131;
then A2: f . c in {0 } by TARSKI:def 1;
c in dom (abs f) by A1, FUNCT_1:def 13;
then c in dom f by VALUED_1:def 11;
hence c in f " {0 } by A2, FUNCT_1:def 13; :: thesis: verum

end;

assume A3: c in f " {0 } ; :: thesis: c in (abs f) " {0 }
then f . c in {0 } by FUNCT_1:def 13;
then f . c = 0 by TARSKI:def 1;
then abs (f . c) = 0 by ABSVALUE:7;
then (abs f) . c = 0 by VALUED_1:18;
then A4: (abs f) . c in {0 } by TARSKI:def 1;
c in dom f by A3, FUNCT_1:def 13;
then c in dom (abs f) by VALUED_1:def 11;
hence c in (abs f) " {0 } by A4, FUNCT_1:def 13; :: thesis: verum

end;

hence (abs f) " {0 } = f " {0 } by TARSKI:2; :: thesis: (- f) " {0 } = f " {0 }

now

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

proof

assume A5: c in (- f) " {0 } ; :: thesis: c in f " {0 }
then (- f) . c in {0 } by FUNCT_1:def 13;
then (- f) . c = 0 by TARSKI:def 1;
then - (- (f . c)) = - 0 by VALUED_1:8;
then A6: f . c in {0 } by TARSKI:def 1;
c in dom (- f) by A5, FUNCT_1:def 13;
then c in dom f by VALUED_1:8;
hence c in f " {0 } by A6, FUNCT_1:def 13; :: thesis: verum

end;

assume A7: c in f " {0 } ; :: thesis: c in (- f) " {0 }
then f . c in {0 } by FUNCT_1:def 13;
then f . c = 0 by TARSKI:def 1;
then (- f) . c = - 0 by VALUED_1:8;
then A8: (- f) . c in {0 } by TARSKI:def 1;
c in dom f by A7, FUNCT_1:def 13;
then c in dom (- f) by VALUED_1:8;
hence c in (- f) " {0 } by A8, FUNCT_1:def 13; :: thesis: verum

end;

hence (- f) " {0 } = f " {0 } by TARSKI:2; :: thesis: verum