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 Th19:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
of S, f be PartFunc of X,ExtREAL, A be Element of S st (ex E be Element of S st
  E = dom f & f is E-measurable) & M.A = 0 holds f|A is_integrable_on M
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, A be Element of S;
  assume that
A1: ex E be Element of S st E = dom f & f is E-measurable and
A2: M.A = 0;
A3: dom f=dom (max+ f) by MESFUNC2:def 2;
  max+f is nonnegative by Lm1;
  then integral+(M,(max+ f)|A)=0 by A1,A2,A3,MESFUNC2:25,MESFUNC5:82;
  then
A4: integral+(M,max+(f|A)) < +infty by MESFUNC5:28;
  consider E be Element of S such that
A5: E = dom f and
A6: f is E-measurable by A1;
A7: dom f /\ (A /\ E) = A /\ E by A5,XBOOLE_1:17,28;
A8: dom f=dom (max- f) by MESFUNC2:def 3;
  max-f is nonnegative by Lm1;
  then integral+(M,(max- f)|A)=0 by A1,A2,A8,MESFUNC2:26,MESFUNC5:82;
  then
A9: integral+(M,max-(f|A)) < +infty by MESFUNC5:28;
A10: dom (f|A) = dom f /\ A by RELAT_1:61;
  f is (A/\E)-measurable by A6,MESFUNC1:30,XBOOLE_1:17;
  then
A11: f|(A /\ E) is (A/\E)-measurable by A7,MESFUNC5:42;
  f|(A /\ E) = f|A /\ f|E by RELAT_1:79
    .= f|A /\ f by A5
    .= f|A by RELAT_1:59,XBOOLE_1:28;
  hence thesis by A5,A11,A10,A4,A9;
end;
