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 Th65:
  for X, A being set, f being Function of X, REAL holds (-f).:A = --(f.:A)
proof
  let X, A be set, f be Function of X, REAL;
  now
    let x be object;
    hereby
      assume 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);
    end;
    assume x in --(f.:A); then
    consider r3 being Complex such that
A2: x = -r3 and
A3: r3 in f.:A;
    reconsider 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;
  end;
  hence thesis by TARSKI:2;
end;
