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 & A misses B implies Integral(M,f|(A\/B)) =
  Integral(M,f|A) + Integral(M,f|B)
proof
  assume that
A1: f is_integrable_on M and
A2: A misses B;
A3: f|B is_integrable_on M by A1,Th23;
  then
A4: Re(f|B) is_integrable_on M;
  then
A5: Integral(M,Re(f|B)) < +infty by MESFUNC6:90;
A6: Im(f|B) is_integrable_on M by A3;
  then
A7: -infty < Integral(M,Im(f|B)) by MESFUNC6:90;
A8: Integral(M,Im(f|B)) < +infty by A6,MESFUNC6:90;
  -infty < Integral(M,Re(f|B)) by A4,MESFUNC6:90;
  then reconsider R2=Integral(M,Re(f|B)), I2=Integral(M,Im(f|B))
as Element of REAL by A5
,A7,A8,XXREAL_0:14;
A9: f|A is_integrable_on M by A1,Th23;
  then
A10: Re(f|A) is_integrable_on M;
  then
A11: Integral(M,Re(f|A)) < +infty by MESFUNC6:90;
  set C=A\/B;
A12: f|(A\/B) is_integrable_on M by A1,Th23;
  then
A13: Re(f|C) is_integrable_on M;
  then
A14: Integral(M,Re(f|C)) < +infty by MESFUNC6:90;
A15: Im(f|C) is_integrable_on M by A12;
  then
A16: -infty < Integral(M,Im(f|C)) by MESFUNC6:90;
A17: Integral(M,Im(f|C)) < +infty by A15,MESFUNC6:90;
  -infty < Integral(M,Re(f|C)) by A13,MESFUNC6:90;
  then reconsider R3=Integral(M,Re(f|C)), I3=Integral(M,Im(f|C))
as Element of REAL
  by A14,A16,A17,XXREAL_0:14;
A18: Integral(M,f|(A\/B)) = R3 + I3 * <i> by A12,Def3;
A19: Im(f|A) is_integrable_on M by A9;
  then
A20: -infty < Integral(M,Im(f|A)) by MESFUNC6:90;
A21: Integral(M,Im(f|A)) < +infty by A19,MESFUNC6:90;
  -infty < Integral(M,Re(f|A)) by A10,MESFUNC6:90;
  then reconsider R1=Integral(M,Re(f|A)), I1=Integral(M,Im(f|A))
as Element of REAL by A11
,A20,A21,XXREAL_0:14;
  Im f is_integrable_on M by A1;
  then
  Integral(M,(Im f)|(A\/B)) = Integral(M,(Im f)|A) + Integral(M,(Im f)|B)
  by A2,MESFUNC6:92;
  then Integral(M,Im(f)|C) = Integral(M,Im(f|A)) + Integral(M,Im(f)|B) by Th7
    .= Integral(M,Im(f|A)) + Integral(M,Im(f|B)) by Th7;
  then I3 = I1 + (I2 qua ExtReal) by Th7;
  then
A22: I3 = I1 + I2;
  Re f is_integrable_on M by A1;
  then
  Integral(M,(Re f)|(A\/B)) = Integral(M,(Re f)|A) + Integral(M,(Re f)|B)
  by A2,MESFUNC6:92;
  then Integral(M,Re(f)|C) = Integral(M,Re(f|A)) + Integral(M,Re(f)|B) by Th7
    .= Integral(M,Re(f|A)) + Integral(M,Re(f|B)) by Th7;
  then R3 = R1 + (R2 qua ExtReal) by Th7;
  then R3 = R1 + R2;
  then Integral(M,f|(A\/B)) = (R1 + I1 * <i>) + (R2 + I2 * <i>) by A22,A18;
  then Integral(M,f|(A\/B)) = Integral(M,f|A) + (R2 + I2 * <i>) by A9,Def3;
  hence thesis by A3,Def3;
end;
