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 & B = (dom f)\A implies f|A is_integrable_on M &
  Integral(M,f) = Integral(M,f|A)+Integral(M,f|B)
proof
  assume that
A1: f is_integrable_on M and
A2: B = (dom f)\A;
A3: Re f is_integrable_on M by A1;
  then
A4: Integral(M,Re f) < +infty by MESFUNC6:90;
A5: Im f is_integrable_on M by A1;
  then
A6: -infty < Integral(M,Im f) by MESFUNC6:90;
A7: Integral(M,Im f) < +infty by A5,MESFUNC6:90;
  -infty < Integral(M,Re f) by A3,MESFUNC6:90;
  then reconsider R=Integral(M,Re f), I=Integral(M,Im f)
as Element of REAL by A4,A6,A7,
XXREAL_0:14;
A8: Integral(M,f) = R + I * <i> by A1,Def3;
A9: f|B is_integrable_on M by A1,Th23;
  then
A10: Re(f|B) is_integrable_on M;
  then
A11: Integral(M,Re(f|B)) < +infty by MESFUNC6:90;
A12: Im(f|B) is_integrable_on M by A9;
  then
A13: -infty < Integral(M,Im(f|B)) by MESFUNC6:90;
A14: Integral(M,Im(f|B)) < +infty by A12,MESFUNC6:90;
  -infty < Integral(M,Re(f|B)) by A10,MESFUNC6:90;
  then reconsider R2=Integral(M,Re(f|B)), I2=Integral(M,Im(f|B))
as Element of REAL by A11
,A13,A14,XXREAL_0:14;
A15: f|A is_integrable_on M by A1,Th23;
  then
A16: Re(f|A) is_integrable_on M;
  then
A17: Integral(M,Re(f|A)) < +infty by MESFUNC6:90;
A18: Im(f|A) is_integrable_on M by A15;
  then
A19: -infty < Integral(M,Im(f|A)) by MESFUNC6:90;
A20: Integral(M,Im(f|A)) < +infty by A18,MESFUNC6:90;
  -infty < Integral(M,Re(f|A)) by A16,MESFUNC6:90;
  then reconsider R1=Integral(M,Re(f|A)), I1=Integral(M,Im(f|A))
as Element of REAL by A17
,A19,A20,XXREAL_0:14;
  dom f = dom Im f by COMSEQ_3:def 4;
  then Integral(M,Im(f)) = Integral(M,Im(f)|A)+Integral(M,Im(f)|B) by A2,A5,
MESFUNC6:93;
  then Integral(M,Im f) = 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
A21: I = I1 + I2 by SUPINF_2:1;
  dom f = dom Re f by COMSEQ_3:def 3;
  then Integral(M,Re(f)) = Integral(M,Re(f)|A)+Integral(M,Re(f)|B) by A2,A3,
MESFUNC6:93;
  then Integral(M,Re f) = 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 R = R1 + R2 by SUPINF_2:1;
  then Integral(M,f) = (R1 + I1 * <i>) + (R2 + I2 * <i>) by A21,A8;
  then Integral(M,f) = Integral(M,f|A) + (R2 + I2 * <i>) by A15,Def3;
  hence thesis by A1,A9,Def3,Th23;
end;
