reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;
reserve r3, r1, q3, p3 for Real;

theorem
  for X being set, A being Subset of REAL, f being Function of X, REAL
    holds (-f)"A = f"(--A)
proof
  let X be set, A be Subset of REAL, f be Function of X, REAL;
  now
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    hereby
A1:   (-f).x = -(f.xx) by VALUED_1:8;
      assume
A2:   x in (-f)"A;
      then (-f).x in A by FUNCT_2:38;
      then - -f.xx in --A by A1;
      hence x in f"(--A) by A2,FUNCT_2:38;
    end;
A3: (-f).x = -(f.xx) by VALUED_1:8;
    assume
A4: x in f"(--A);
    then f.x in --A by FUNCT_2:38;
    then (-f).x in -- --A by A3;
    hence x in (-f)"A by A4,FUNCT_2:38;
  end;
  hence thesis by TARSKI:2;
end;
