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
  f is_integrable_on M & g is_integrable_on M & B c= dom(f+g) implies f+
g is_integrable_on M & Integral_on(M,B,f+g) = Integral_on(M,B,f) + Integral_on(
  M,B,g)
proof
  assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M and
A3: B c= dom(f+g);
A4: Re f is_integrable_on M by A1;
  then Re(f)|B is_integrable_on M by MESFUNC6:91;
  then
A5: Re(f|B) is_integrable_on M by Th7;
  then
A6: Integral(M,Re(f|B)) < +infty by MESFUNC6:90;
A7: Im f is_integrable_on M by A1;
  then Im(f)|B is_integrable_on M by MESFUNC6:91;
  then
A8: Im(f|B) is_integrable_on M by Th7;
  then
A9: -infty < Integral(M,Im(f|B)) by MESFUNC6:90;
A10: Integral(M,Im(f|B)) < +infty by A8,MESFUNC6:90;
  -infty < Integral(M,Re(f|B)) by A5,MESFUNC6:90;
  then reconsider R1=Integral(M,Re(f|B)), I1=Integral(M,Im(f|B))
as Element of REAL by A6
,A9,A10,XXREAL_0:14;
  f|B is_integrable_on M by A5,A8;
  then
A11: Integral(M,f|B) = R1 + I1*<i> by Def3;
A12: Im g is_integrable_on M by A2;
  then Im(g)|B is_integrable_on M by MESFUNC6:91;
  then
A13: Im(g|B) is_integrable_on M by Th7;
  then
A14: -infty < Integral(M,Im(g|B)) by MESFUNC6:90;
A15: Re g is_integrable_on M by A2;
  then Re(g)|B is_integrable_on M by MESFUNC6:91;
  then
A16: Re(g|B) is_integrable_on M by Th7;
  then
A17: Integral(M,Re(g|B)) < +infty by MESFUNC6:90;
  B c= dom Im(f+g) by A3,COMSEQ_3:def 4;
  then
A18: B c= dom(Im(f)+Im(g)) by Th5;
  then
  Integral_on(M,B,Im(f)+Im(g)) = Integral_on(M,B,Im f) + Integral_on(M,B,
  Im g) by A7,A12,MESFUNC6:103;
  then
A19: Integral(M,Im(f+g)|B) = Integral(M,Im(f)|B) + Integral(M,Im(g)|B) by Th5;
A20: Integral(M,Im(g|B)) < +infty by A13,MESFUNC6:90;
  -infty < Integral(M,Re(g|B)) by A16,MESFUNC6:90;
  then reconsider R2=Integral(M,Re(g|B)), I2=Integral(M,Im(g|B))
as Element of REAL by A17
,A14,A20,XXREAL_0:14;
  g|B is_integrable_on M by A16,A13;
  then
A21: Integral(M,g|B) = R2 + I2*<i> by Def3;
A22: Integral(M,Im(g)|B) = I2 by Th7;
  Im(f)+Im(g) is_integrable_on M by A7,A12,A18,MESFUNC6:103;
  then
A23: Im(f+g) is_integrable_on M by Th5;
  then Im(f+g)|B is_integrable_on M by MESFUNC6:91;
  then
A24: Im((f+g)|B) is_integrable_on M by Th7;
  then
A25: -infty < Integral(M,Im((f+g)|B)) by MESFUNC6:90;
  B c= dom Re(f+g) by A3,COMSEQ_3:def 3;
  then
A26: B c= dom(Re(f)+Re(g)) by Th5;
  then
  Integral_on(M,B,Re(f)+Re(g)) = Integral_on(M,B,Re f) + Integral_on(M,B,
  Re g) by A4,A15,MESFUNC6:103;
  then
A27: Integral(M,Re(f+g)|B) = Integral(M,Re(f)|B) + Integral(M,Re(g)|B) by Th5;
  Re(f)+Re(g) is_integrable_on M by A4,A15,A26,MESFUNC6:103;
  then
A28: Re(f+g) is_integrable_on M by Th5;
  then Re(f+g)|B is_integrable_on M by MESFUNC6:91;
  then
A29: Re((f+g)|B) is_integrable_on M by Th7;
  then
A30: Integral(M,Re((f+g)|B)) < +infty by MESFUNC6:90;
A31: Integral(M,Im((f+g)|B)) < +infty by A24,MESFUNC6:90;
  -infty < Integral(M,Re((f+g)|B)) by A29,MESFUNC6:90;
  then reconsider
  R=Integral(M,Re((f+g)|B)), I=Integral(M,Im((f+g)|B))
as Element of REAL by A30,A25,A31,
XXREAL_0:14;
A32: Integral(M,Re(g)|B) = R2 by Th7;
  Integral(M,Im(f)|B) = I1 by Th7;
  then I = I1 + (I2 qua ExtReal) by A19,A22,Th7;
  then
A33: I = I1 + I2;
  Integral(M,Re(f)|B) = R1 by Th7;
  then R = R1 + (R2 qua ExtReal) by A27,A32,Th7;
  then
A34: R = R1 + R2;
  (f+g)|B is_integrable_on M by A29,A24;
  then Integral_on(M,B,f+g) = R + I * <i> by Def3
    .= Integral_on(M,B,f) + Integral_on(M,B,g) by A34,A33,A11,A21;
  hence thesis by A28,A23;
end;
