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, q3 holds (q3+f)"A = f"(-q3++A)
proof
  let X be set, A be Subset of REAL, f be Function of X, REAL,
      q3 be Real;
  now
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    hereby
      assume
A1:   x in (q3+f)"A;
      then (q3+f).x in A & (q3+f).x = q3+(f.xx) by FUNCT_2:38,VALUED_1:2;
      then -q3+(q3+(f.xx)) in { -q3 + p3 : p3 in A };
      then -q3+(q3+(f.xx)) in -q3++A by Lm5;
      hence x in f"(-q3++A) by A1,FUNCT_2:38;
    end;
    assume
A2: x in f"(-q3++A);
    then f.x in -q3++A & (q3+f).x = q3+(f.xx) by FUNCT_2:38,VALUED_1:2;
    then (q3+f).x in { q3+p3 : p3 in -q3 ++ A };
    then (q3+f).x in q3++(-q3++A) by Lm5;
    then (q3+f).x in (q3+-q3)++A by MEMBER_1:147;
    then (q3+f).x in A by MEMBER_1:146;
    hence x in (q3+f)"A by A2,FUNCT_2:38;
  end;
  hence thesis by TARSKI:2;
end;
