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 without_zero Subset of REAL,
      f being Function of X, REAL holds (Inv f)"A = f"(Inv A)
proof
  let X be set, A be without_zero Subset of REAL, f be Function of X, REAL;
  now
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    hereby
A1:   (Inv f).x = (f.xx)" by VALUED_1:10;
      assume
A2:   x in (Inv f)"A;
      then (Inv f).x in A by FUNCT_2:38;
      then 1/((f.xx)") in Inv A by A1;
      hence x in f"(Inv A) by A2,FUNCT_2:38;
    end;
A3: (f.xx)" = 1/(f.xx) & (Inv f).x = (f.xx)" by VALUED_1:10;
    assume
A4: x in f"(Inv A);
    then f.x in Inv A by FUNCT_2:38;
    then (Inv f).x in Inv Inv A by A3;
    hence x in (Inv f)"A by A4,FUNCT_2:38;
  end;
  hence thesis by TARSKI:2;
end;
