reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_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 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;

theorem Th22:
  for X,S,M for f be PartFunc of X,ExtREAL st (ex E st E = dom f)
& for x st x in dom f holds 0= f.x holds f is_integrable_on M & Integral(M,f) =
  0
proof
  let X,S,M;
  let f be PartFunc of X,ExtREAL;
  given E such that
A1: E = dom f;
  assume
A2: for x st x in dom f holds 0 = f.x;
  then
A3: for x be object st x in dom f holds f.x = 0;
  then
A4: f is_simple_func_in S by A1,MESFUNC6:2;
  then
A5: integral+(M,f) = 0 by A1,A2,MESFUNC2:34,MESFUNC5:87;
A6: dom max+f = dom f by MESFUNC2:def 2;
A7: now
    let x be Element of X;
    assume
A8: x in dom f;
    hence f.x = max(0,0) by A2
      .=max(f.x,0) by A2,A8
      .= (max+f).x by A6,A8,MESFUNC2:def 2;
  end;
A9: f is E-measurable by A1,A3,MESFUNC2:34,MESFUNC6:2;
A10: f is nonnegative
  proof
    let y be ExtReal;
    assume y in rng f;
    then ex x1 being object st x1 in dom f & y = f.x1 by FUNCT_1:def 3;
    hence y >= 0 by A2;
  end;
  then 0. = Integral(M,f) by A1,A4,A5,MESFUNC2:34,MESFUNC5:88
    .= integral+(M,f)-integral+(M,max-f) by A6,A7,PARTFUN1:5
    .= 0.+(-integral+(M,max-f)) by A1,A2,A4,MESFUNC2:34,MESFUNC5:87
    .= -integral+(M,max-f) by XXREAL_3:4;
  then
A11: --integral+(M,max-f) = -0;
  integral+(M,max+f) < +infty by A6,A7,A5,PARTFUN1:5;
  hence f is_integrable_on M by A1,A9,A11;
  thus thesis by A1,A9,A5,A10,MESFUNC5:88;
end;
