
theorem Th27:
  for C being non empty set, f being PartFunc of C,ExtREAL, c be
  Real st 0 <= c holds max+((-c)(#)f) = c(#)max-f & max-((-c)(#)f) = c(#)max+f
proof
  let C be non empty set;
  let f be PartFunc of C,ExtREAL;
  let c be Real;
  assume
A1: 0 <= c;
A2: dom max+((-c)(#)f) = dom((-c)(#)f) by MESFUNC2:def 2;
  then dom max+((-c)(#)f) = dom f by MESFUNC1:def 6;
  then
A3: dom max+((-c)(#)f) = dom max-f by MESFUNC2:def 3;
  then
A4: dom max+((-c)(#)f) = dom(c(#)max-f) by MESFUNC1:def 6;
  for x be Element of C st x in dom max+((-c)(#)f) holds max+((-c)(#)f).x
  = (c(#)max-f).x
  proof
    let x be Element of C;
    assume
A5: x in dom max+((-c)(#)f);
    then
A6: (max+((-c)(#)f)).x = max(((-c)(#)f).x,0) by MESFUNC2:def 2
      .= max((-c)*f.x,0) by A2,A5,MESFUNC1:def 6
      .= max(-( c * f.x),0) by XXREAL_3:92;
    (c(#)max-f).x =  c * max-f.x by A4,A5,MESFUNC1:def 6
      .=  c * max(-f.x, 0) by A3,A5,MESFUNC2:def 3
      .= max( c * (-f.x), c *  0) by A1,Th6
      .= max(-( c * f.x), c * (0 qua ExtReal)) by XXREAL_3:92;
    hence thesis by A6;
  end;
  hence max+((-c)(#)f) = c(#)max-f by A4,PARTFUN1:5;
A7: dom max-((-c)(#)f) = dom((-c)(#)f) by MESFUNC2:def 3;
  then dom max-((-c)(#)f) = dom f by MESFUNC1:def 6;
  then
A8: dom max-((-c)(#)f) = dom max+f by MESFUNC2:def 2;
  then
A9: dom max-((-c)(#)f) = dom(c(#) max+ f) by MESFUNC1:def 6;
  for x be Element of C st x in dom max-((-c)(#)f) holds max-((-c)(#)f).x
  = (c(#)max+f).x
  proof
    let x be Element of C;
    assume
A10: x in dom max-((-c)(#)f);
    then
A11: max-((-c)(#)f).x = max(-((-c)(#)f).x,0) by MESFUNC2:def 3
      .= max(-((-c))*f.x,0) by A7,A10,MESFUNC1:def 6
      .= max((-(-( c)))*f.x,0) by XXREAL_3:92;
    (c(#)max+f).x =  c * max+f.x by A9,A10,MESFUNC1:def 6
      .=  c * max(f.x, 0) by A8,A10,MESFUNC2:def 2
      .= max( c * f.x, c * (0 qua ExtReal)) by A1,Th6;
    hence thesis by A11;
  end;
  hence thesis by A9,PARTFUN1:5;
end;
