
theorem Th61:
  for X be non empty set, S be SigmaField of X,
  f,g be without+infty PartFunc of X,ExtREAL, A be Element of S 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,
   f,g be without+infty PartFunc of X,ExtREAL,
   A be Element of S;
   assume that
A3: f is A-measurable and
A4: g is A-measurable and
A5: A c= dom(f+g);
A6:dom(f+g) = dom f /\ dom g by MESFUNC9:1;
   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 A5,A6; then
   -f is A-measurable & -g is A-measurable by A3,A4,Th59; then
A7:(-f)+(-g) is A-measurable by MESFUNC5:31;
   dom f = dom(-f) & dom g = dom(-g) by MESFUNC1:def 7; then
   dom(-f + -g) = dom f /\ dom g by MESFUNC5:16 .= dom(f+g) by MESFUNC9:1; then
A8:-((-f)+(-g)) is A-measurable by A5,A7,Th59;
   (-f)+(-g) = -(f+g) by Th60;
   hence thesis by A8,DBLSEQ_3:2;
end;
