let X, A be set ; :: thesis: for f being Function of X,REAL holds (- f) .: A = -- (f .: A)
let f be Function of X,REAL; :: thesis: (- f) .: A = -- (f .: A)

now :: thesis: for x being object holds
( ( x in (- f) .: A implies x in -- (f .: A) ) & ( x in -- (f .: A) implies x in (- f) .: A ) )

let x be object ; :: thesis: ( ( x in (- f) .: A implies x in -- (f .: A) ) & ( x in -- (f .: A) implies x in (- f) .: A ) )

hereby :: thesis: ( x in -- (f .: A) implies x in (- f) .: A )

assume x in (- f) .: A ; :: thesis: x in -- (f .: A)
then consider x1 being object such that
A1: ( x1 in X & x1 in A & x = (- f) . x1 ) by FUNCT_2:64;
( x = - (f . x1) & f . x1 in f .: A ) by A1, FUNCT_2:35, VALUED_1:8;
hence x in -- (f .: A) ; :: thesis: verum

end;

assume x in -- (f .: A) ; :: thesis: x in (- f) .: A
then consider r3 being Complex such that
A2: x = - r3 and
A3: r3 in f .: A ;
reconsider r3 = r3 as Real by A3;
consider x1 being object such that
A4: ( x1 in X & x1 in A ) and
A5: r3 = f . x1 by A3, FUNCT_2:64;
x = (- f) . x1 by A2, A5, VALUED_1:8;
hence x in (- f) .: A by A4, FUNCT_2:35; :: thesis: verum

end;

hence (- f) .: A = -- (f .: A) by TARSKI:2; :: thesis: verum