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 Th36:
  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;
A3: Re g is_integrable_on M by A2;
  Re f is_integrable_on M by A1;
  then consider E be Element of S such that
A4: E = dom Re f /\ dom Re g and
A5: Integral(M,Re(f)+Re(g))=Integral(M,(Re f)|E)+Integral(M,(Re g)|E) by A3,
MESFUNC6:101;
A6: f|E is_integrable_on M by A1,Th23;
  then
A7: Re(f|E) is_integrable_on M;
  then
A8: Integral(M,Re(f|E)) < +infty by MESFUNC6:90;
A9: Im(f|E) is_integrable_on M by A6;
  then
A10: -infty < Integral(M,Im(f|E)) by MESFUNC6:90;
A11: Integral(M,Im(f|E)) < +infty by A9,MESFUNC6:90;
  -infty < Integral(M,Re(f|E)) by A7,MESFUNC6:90;
  then reconsider R1=Integral(M,Re(f|E)), I1=Integral(M,Im(f|E))
as Element of REAL by A8
,A10,A11,XXREAL_0:14;
A12: dom f = dom Re f by COMSEQ_3:def 3;
A13: Im g is_integrable_on M by A2;
  Im f is_integrable_on M by A1;
  then
A14: ex Ei be Element of S st Ei = dom Im f /\ dom Im g & Integral(M,Im(f)+Im
  (g))=Integral(M,Im(f)|Ei)+Integral(M,Im(g)|Ei) by A13,MESFUNC6:101;
A15: Integral(M,Im(f)+Im(g)) = Integral(M,Im(f+g)) by Th5;
A16: f+g is_integrable_on M by A1,A2,Th33;
  then
A17: Re(f+g) is_integrable_on M;
  then
A18: Integral(M,Re(f+g)) < +infty by MESFUNC6:90;
A19: Im(f+g) is_integrable_on M by A16;
  then
A20: -infty < Integral(M,Im(f+g)) by MESFUNC6:90;
A21: Integral(M,Im(f+g)) < +infty by A19,MESFUNC6:90;
  -infty < Integral(M,Re(f+g)) by A17,MESFUNC6:90;
  then reconsider R=Integral(M,Re(f+g)), I=Integral(M,Im(f+g))
as Element of REAL by A18
,A20,A21,XXREAL_0:14;
A22: Integral(M,f+g) = R + I * <i> by A16,Def3;
A23: dom g = dom Im g by COMSEQ_3:def 4;
A24: g|E is_integrable_on M by A2,Th23;
  then
A25: Re(g|E) is_integrable_on M;
  then
A26: Integral(M,Re(g|E)) < +infty by MESFUNC6:90;
A27: Im(g|E) is_integrable_on M by A24;
  then
A28: -infty < Integral(M,Im(g|E)) by MESFUNC6:90;
A29: Integral(M,Im(g|E)) < +infty by A27,MESFUNC6:90;
  -infty < Integral(M,Re(g|E)) by A25,MESFUNC6:90;
  then reconsider R2=Integral(M,Re(g|E)), I2=Integral(M,Im(g|E))
as Element of REAL by A26
,A28,A29,XXREAL_0:14;
A30: dom g = dom Re g by COMSEQ_3:def 3;
  Integral(M,Re(f)+Re(g)) = Integral(M,Re(f+g)) by Th5;
  then Integral(M,Re(f+g)) = Integral(M,Re(f|E)) + Integral(M,Re(g)|E) by A5
,Th7
    .= Integral(M,Re(f|E)) + Integral(M,Re(g|E)) by Th7;
  then
A31: R = R1 + R2 by SUPINF_2:1;
  dom f = dom Im f by COMSEQ_3:def 4;
  then Integral(M,Im(f+g)) = Integral(M,Im(f|E)) + Integral(M,Im(g)|E) by A4
,A12,A30,A23,A14,A15,Th7;
  then I = I1 + (I2 qua ExtReal) by Th7;
  then I = I1 + I2;
  then Integral(M,f+g) = (R1 + I1 * <i>) + (R2 + I2 * <i>) by A31,A22;
  then Integral(M,f+g) = Integral(M,f|E) + (R2 + I2 * <i>) by A6,Def3;
  then Integral(M,f+g) = Integral(M,f|E)+Integral(M,g|E) by A24,Def3;
  hence thesis by A4,A12,A30;
end;
