reserve a,b,r,x,y for Real,
  i,j,k,n for Nat,
  x1 for set;

theorem Th17:
  for X be non empty set, f be Function of X,REAL holds rng(r(#)f)
  = r**rng f
proof
  let X be non empty set;
  let f be Function of X,REAL;
  for y being Element of REAL holds y in r**rng f implies y in rng(r(#)f)
  proof
    let y be Element of REAL;
    assume y in r**rng f;
    then y in {r*b : b in rng f} by Th8;
    then consider b such that
A1: y=r*b and
A2: b in rng f;
    consider x being Element of X such that
A3: x in dom f and
A4: b=f.x by A2,PARTFUN1:3;
    x in dom(r(#)f) by A3,VALUED_1:def 5;
    then (r(#)f).x in rng(r(#)f) by FUNCT_1:def 3;
    hence thesis by A1,A4,RFUNCT_1:57;
  end;
  then
A5: r**rng f c= rng(r(#)f);
  for y being Element of REAL holds y in rng(r(#)f) implies y in r**rng f
  proof
    let y be Element of REAL;
    assume y in rng(r(#)f);
    then consider x being Element of X such that
A6: x in dom(r(#)f) and
A7: y=(r(#)f).x by PARTFUN1:3;
    x in dom f by A6,VALUED_1:def 5;
    then
A8: f.x in rng f by FUNCT_1:def 3;
    reconsider fx=f.x as Real;
    y = r*(fx) by A7,RFUNCT_1:57;
    then y in {r*b : b in rng f} by A8;
    hence thesis by Th8;
  end;
  then rng(r(#)f) c= r**rng f;
  hence thesis by A5;
end;
