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
    .= dom f by VALUED_1:def 5
    .= dom max+f by RFUNCT_3:def 10
    .= 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
A3: x in dom max+(r(#)f);
    then
A4: x in dom(r(#)f) by RFUNCT_3:def 10;
    then x in dom f by VALUED_1:def 5;
    then
A5: x in dom max+ f by RFUNCT_3:def 10;
A6: (max+(r(#)f)).x = max+((r(#)f).x) by A3,RFUNCT_3:def 10
      .= max(r*f.x,0) by A4,VALUED_1:def 5;
    (r(#)max+f).x = r * max+f.x by A2,A3,VALUED_1:def 5
      .= rr * max+(f.x) by A5,RFUNCT_3:def 10
      .= max(rr * f.x,rr * 0) by A1,FUZZY_2:41;
    hence thesis by A6;
  end;
  hence max+(r(#)f) = r(#)max+f by A2,PARTFUN1:5;
A7: dom(max-(r(#)f)) = dom(r(#)f) by RFUNCT_3:def 11
    .= dom f by VALUED_1:def 5
    .= dom max-f by RFUNCT_3:def 11
    .= 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
A8: x in dom max-(r(#)f);
    then
A9: x in dom(r(#)f) by RFUNCT_3:def 11;
    then x in dom f by VALUED_1:def 5;
    then
A10: x in dom max- f by RFUNCT_3:def 11;
A11: (max-(r(#)f)).x = max-((r(#)f).x) by A8,RFUNCT_3:def 11
      .= max(-(r*f.x),0) by A9,VALUED_1:def 5;
    (r(#)max-f).x = r * max-f.x by A7,A8,VALUED_1:def 5
      .= rr * max-(f.x) by A10,RFUNCT_3:def 11
      .= max(rr*(-(f.x qua Real)qua Real),rr*0 )
              by A1,FUZZY_2:41
      .= max(-(r)*f.x,r*0);
    hence thesis by A11;
  end;
  hence thesis by A7,PARTFUN1:5;
end;
