
theorem
  for X be non empty set, S be SigmaField of X, A be Element of S,
  f be without-infty PartFunc of X,ExtREAL,
  g be without+infty PartFunc of X,ExtREAL
 st f is A-measurable & g is A-measurable & A c= dom(f-g) holds
  f-g is A-measurable
proof
   let X be non empty set, S be SigmaField of X, A be Element of S,
   f be without-infty PartFunc of X,ExtREAL,
   g be without+infty PartFunc of X,ExtREAL;
   assume that
A1: f is A-measurable and
A2: g is A-measurable and
A3: A c= dom(f-g);
A4:dom(f-g) = dom f /\ dom g by MESFUNC5:17;
   dom(-f+g) = dom(-(f-g)) by Th60; then
A5:dom(-f+g) = dom(f-g) by MESFUNC1:def 7;
   dom f /\ dom g c= dom f & dom f /\ dom g c= dom g by XBOOLE_1:17; then
   A c= dom f & A c= dom g by A3,A4; then
   -f is A-measurable by A1,Th59; then
A6:(-f)+g is A-measurable by A2,A3,A5,Th61;
   dom f = dom(-f) & dom g = dom(-g) by MESFUNC1:def 7; then
   dom(-f + g) = dom f /\ dom g by MESFUNC9:1; then
   dom(-f + g) = dom(f-g) by MESFUNC5:17; then
   -((-f)+g) is A-measurable by A3,A6,Th59;
   hence f-g is A-measurable by Th60;
end;
