 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);
reserve u,v,w for VECTOR of CLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2,A,B for Element of S,
  f,g,h,f1,g1 for PartFunc of X,COMPLEX;

theorem Th16:
  for f be PartFunc of X,COMPLEX st dom f in S
  & for x st x in dom f holds 0= f.x holds
  f is_integrable_on M & Integral(M,f) = 0
proof
  let f be PartFunc of X,COMPLEX;
  assume  dom f in S; then
  reconsider E = dom f as Element of S;
  assume
A1: for x st x in dom f holds 0 = f.x;
A2: for x st x in dom f holds (Re f).x = 0 & (Im f).x = 0
proof
let x such that
A3: x in dom f;
A4: x in dom (Re f) & x in dom(Im f) by A3,COMSEQ_3:def 3,def 4;
A5: (Re f).x = Re (f.x) by COMSEQ_3:def 3,A4
            .= 0 by A3,A1,COMPLEX1:4;
    (Im f).x = Im (f.x) by COMSEQ_3:def 4,A4
            .= 0 by A3,A1,COMPLEX1:4;
hence thesis by A5;
end;
A6: dom Re f = E by COMSEQ_3:def 3;
A7: dom Im f = E by COMSEQ_3:def 4;
set f1 = Re f;
A8:  E=dom f1 & for x st x in dom f1 holds 0 = f1.x by A2,A6;
A9: R_EAL f1 is_integrable_on M & Integral(M,R_EAL f1) = 0 by LPSPACE1:22,A8;
A10:  Re f is_integrable_on M & Integral(M,Re f) = 0 by A9;
     set f2 = Im f;
A11: E=dom f2 & for x st x in dom f2 holds 0 = f2.x by A2,A7;
A12: R_EAL f2 is_integrable_on M & Integral(M,R_EAL f2) = 0 by LPSPACE1:22,A11;
A13: Im f is_integrable_on M & Integral(M,Im f) = 0 by A12;
     f is_integrable_on M by A10,A13,MESFUN6C:def 2; then
     ex RF, IF be Real st
     RF = Integral(M,Re f) & IF= Integral(M,Im f) &
     Integral(M,f) = RF+IF*<i> by MESFUN6C:def 3;
     hence thesis by A10,A13,MESFUN6C:def 2;
end;
