reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;

theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f,g be PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
  ex E be Element of S st E = dom f /\ dom g & Integral(M,f-g)=Integral(M,f|E)+
  Integral(M,(-g)|E)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g
  be PartFunc of X,REAL;
  assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
  R_EAL g is_integrable_on M by A2;
  then (-jj)(#)R_EAL g is_integrable_on M by MESFUNC5:110;
  then -R_EAL g is_integrable_on M by MESFUNC2:9;
  then
A3: R_EAL -g is_integrable_on M by MESFUNC6:28;
  R_EAL f is_integrable_on M by A1;
  then consider E be Element of S such that
A4: E = dom(R_EAL f) /\ dom(R_EAL -g) and
A5: Integral(M,R_EAL f + R_EAL -g) = Integral(M,(R_EAL f)|E) + Integral(
  M,(R_EAL -g)|E) by A3,MESFUNC5:109;
  take E;
  dom(R_EAL -g) = dom(-R_EAL g) by MESFUNC6:28;
  hence thesis by A4,A5,MESFUNC1:def 7,MESFUNC6:23;
end;
