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

theorem Th18:
  for X,Z be non empty set, f be PartFunc of X,REAL holds rng(r(#)
  (f|Z)) = r**rng(f|Z)
proof
  let X,Z be non empty set;
  let f be PartFunc of X,REAL;
  for y being Element of REAL holds y in r**rng(f|Z) implies y in rng(r(#)f|Z)
  proof
    let y be Element of REAL;
    assume y in r**rng(f|Z);
    then y in {r*b : b in rng(f|Z)} by Th8;
    then consider b such that
A1: y=r*b and
A2: b in rng(f|Z);
    consider x being Element of X such that
A3: x in dom(f|Z) and
A4: b=(f|Z).x by A2,PARTFUN1:3;
A5: x in dom(r(#)f|Z) by A3,VALUED_1:def 5;
    then y= (r(#)f|Z).x by A1,A4,VALUED_1:def 5;
    hence thesis by A5,FUNCT_1:def 3;
  end;
  then
A6: r**rng(f|Z) c= rng(r(#)f|Z);
  for y being Element of REAL holds y in rng(r(#)f|Z) implies y in r**rng(f|Z)
  proof
    let y be Element of REAL;
    assume y in rng(r(#)f|Z);
    then consider x be Element of X such that
A7: x in dom(r(#)f|Z) and
A8: y=(r(#)f|Z).x by PARTFUN1:3;
    x in dom(f|Z) by A7,VALUED_1:def 5;
    then
A9: (f|Z).x in rng(f|Z) by FUNCT_1:def 3;
   reconsider fx = (f|Z).x as Real;
    y=r*fx by A7,A8,VALUED_1:def 5;
    then y in {r*b : b in rng(f|Z)} by A9;
    hence thesis by Th8;
  end;
  then rng(r(#)f|Z) c= r**rng(f|Z);
  hence thesis by A6;
end;
