reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;

theorem
  0 <= r implies max+((-r)(#)f) = r(#)max-f & max-((-r)(#)f) = r(#)max+f
proof
  assume
A1: 0 <= r;
A2: dom max+((-r)(#)f) = dom((-r)(#)f) by RFUNCT_3:def 10;
  then dom max+((-r)(#)f) = dom f by VALUED_1:def 5;
  then
A3: dom max+((-r)(#)f) = dom max-f by RFUNCT_3:def 11;
  then
A4: dom max+((-r)(#)f) = dom(r(#)max-f) by VALUED_1:def 5;
 reconsider rr=r as Real;
  for x be Element of X st x in dom max+((-r)(#)f) holds max+((-r)(#)f).x
  = (r(#)max-f).x
  proof
    let x be Element of X;
    assume
A5: x in dom max+((-r)(#)f);
    then
A6: (max+((-r)(#)f)).x = max+(((-r)(#)f).x) by RFUNCT_3:def 10
      .= max((-r)*f.x,0) by A2,A5,VALUED_1:def 5;
    (r(#)max-f).x = r * max-f.x by A4,A5,VALUED_1:def 5
      .= rr * max-(f.x) by A3,A5,RFUNCT_3:def 11
      .= max(rr * (-(f.x qua Real)qua Real),rr * 0)
          by A1,FUZZY_2:41;
    hence thesis by A6;
  end;
  hence max+((-r)(#)f) = r(#)max-f by A4,PARTFUN1:5;
A7: dom max-((-r)(#)f) = dom((-r)(#)f) by RFUNCT_3:def 11;
  then dom max-((-r)(#)f) = dom f by VALUED_1:def 5;
  then
A8: dom max-((-r)(#)f) = dom max+f by RFUNCT_3:def 10;
  then
A9: dom max-((-r)(#)f) = dom(r(#) max+ f) by VALUED_1:def 5;
  for x be Element of X st x in dom max-((-r)(#)f) holds max-((-r)(#)f).x
  = (r(#)max+f).x
  proof
    let x be Element of X;
    assume
A10: x in dom max-((-r)(#)f);
    then
A11: max-((-r)(#)f).x = max-(((-r)(#)f).x) by RFUNCT_3:def 11
      .= max(-((-r))*f.x,0) by A7,A10,VALUED_1:def 5;
    (r(#)max+f).x = r * max+f.x by A9,A10,VALUED_1:def 5
      .= rr * max+(f.x) by A8,A10,RFUNCT_3:def 10
      .= max(rr*f.x,rr*0) by A1,FUZZY_2:41;
    hence thesis by A11;
  end;
  hence thesis by A9,PARTFUN1:5;
end;
