reserve X for non empty set,
  F for with_the_same_dom Functional_Sequence of X, ExtREAL,
  seq,seq1,seq2 for ExtREAL_sequence,
  x for Element of X,
  a,r for R_eal,
  n,m,k for Nat;
reserve S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S;
reserve F1,F2 for Functional_Sequence of X,ExtREAL,
  f,g,P for PartFunc of X, ExtREAL;

theorem Th13:
  f is_integrable_on M & g is_integrable_on M implies 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
  assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
  (-jj)(#)g is_integrable_on M by A2,MESFUNC5:110;
  then -g is_integrable_on M by MESFUNC2:9;
  then consider E be Element of S such that
A3: E = dom f /\ dom(-g) and
A4: Integral(M,f+(-g))= Integral(M,f|E)+Integral(M,(-g)|E) by A1,MESFUNC5:109;
A5: dom g = dom(-g) by MESFUNC1:def 7;
  Integral(M,f-g)= Integral(M,f|E)+Integral(M,(-g)|E) by A4,MESFUNC2:8;
  hence thesis by A3,A5;
end;
