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
  E = dom f & f is_integrable_on M & M.A =0 implies Integral(M,f|(E\A))
  = Integral(M,f)
proof
  assume that
A1: E = dom f and
A2: f is_integrable_on M and
A3: M.A =0;
  set C=E\A;
A4: dom f = dom Re f by COMSEQ_3:def 3;
A5: Im f is_integrable_on M by A2;
  then R_EAL(Im f) is_integrable_on M;
  then consider IE be Element of S such that
A6: IE = dom(R_EAL(Im f)) and
A7: R_EAL(Im f) is IE-measurable;
A8: dom f = dom Im f by COMSEQ_3:def 4;
A9: Integral(M,Im(f)|C) = Integral(M,Im(f|C)) by Th7;
  Im f is IE-measurable by A7;
  then
A10: Integral(M,Im(f|C)) = Integral(M,Im f) by A1,A3,A8,A6,A9,MESFUNC6:89;
  (Im f)|C is_integrable_on M by A5,MESFUNC6:91;
  then
A11: Im(f|C) is_integrable_on M by Th7;
  then
A12: -infty < Integral(M,Im(f|C)) by MESFUNC6:90;
A13: Re f is_integrable_on M by A2;
  then R_EAL(Re f) is_integrable_on M;
  then consider RE be Element of S such that
A14: RE = dom(R_EAL(Re f)) and
A15: R_EAL(Re f) is RE-measurable;
A16: Integral(M,Re(f)|C) = Integral(M,Re(f|C)) by Th7;
  Re f is RE-measurable by A15;
  then
A17: Integral(M,Re(f|C)) = Integral(M,Re f) by A1,A3,A4,A14,A16,MESFUNC6:89;
  (Re f)|C is_integrable_on M by A13,MESFUNC6:91;
  then
A18: Re(f|C) is_integrable_on M by Th7;
  then
A19: Integral(M,Re(f|C)) < +infty by MESFUNC6:90;
A20: Integral(M,Im(f|C)) < +infty by A11,MESFUNC6:90;
  -infty < Integral(M,Re(f|C)) by A18,MESFUNC6:90;
  then reconsider R2=Integral(M,Re(f|C)), I2=Integral(M,Im(f|C))
as Element of REAL by A19
,A12,A20,XXREAL_0:14;
  f|(E\A) is_integrable_on M by A18,A11;
  hence Integral(M,f|(E\A)) = R2 + I2 * <i> by Def3
    .= Integral(M,f) by A2,A17,A10,Def3;
end;
