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
  (ex E be Element of S st E = dom f & f is E-measurable) & M.A = 0
  implies f|A is_integrable_on M & Integral(M,f|A) = 0
proof
  set g = f|A;
  assume that
A1: ex E be Element of S st E = dom f & f is E-measurable and
A2: M.A = 0;
  consider E be Element of S such that
A3: E = dom f and
A4: f is E-measurable by A1;
A5: dom Im f = dom f by COMSEQ_3:def 4;
A6: Im f is E-measurable by A4;
  then
A7: Integral(M,Im(f)|A)=0 by A2,A3,A5,MESFUNC6:88;
  (Im f)|A is_integrable_on M by A2,A3,A6,A5,Th20;
  then
A8: Im g is_integrable_on M by Th7;
A9: dom Re f = dom f by COMSEQ_3:def 3;
A10: Im g = (Im f)|A by Th7;
A11: Re f is E-measurable by A4;
A12: Re g = (Re f)|A by Th7;
  then reconsider
  R=Integral(M,Re g), I=Integral(M,Im g) as Real by A2,A3,A11,A6,A9,A5,A10,
MESFUNC6:88;
  (Re f)|A is_integrable_on M by A2,A3,A11,A9,Th20;
  then Re g is_integrable_on M by Th7;
  hence g is_integrable_on M by A8;
  hence Integral(M,g) = R + I * <i> by Def3
    .= 0 by A2,A3,A11,A9,A7,A12,A10,MESFUNC6:88;
end;
