 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;
reserve v,u for VECTOR of CLSp_L1Funct M;
reserve v,u for VECTOR of CLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-CSpace M;
reserve x,y for Point of L-1-CSpace M;

theorem
  f in L1_CFunctions M & Integral(M,abs f) = 0 implies
  f a.e.cpfunc= X-->0c,M
proof
  assume that
A1: f in L1_CFunctions M and
A2: Integral(M,abs f) = 0;
A3: ex f1 be PartFunc of X,COMPLEX st f=f1 & ex ND be Element of S st M.ND=0 &
  dom f1 = ND` & f1 is_integrable_on M by A1;
  then consider NDf be Element of S such that
A4: M.NDf=0 and
A5: dom f = NDf` and f is_integrable_on M;
    dom abs f = NDf` & abs f is_integrable_on M
    by A3,A5,Th37,VALUED_1:def 11;
    then
A6: abs f in L1_Functions M by A4;
A7: Integral(M,abs(abs f)) = 0 by A2;
     consider Ef being Element of S such that
A8: M.Ef=0 and
A9: (abs f)|Ef` = (X-->0)|Ef` by A6,A7,LPSPACE1:53,LPSPACE1:def 10;
A10:dom (X-->0) = X by FUNCOP_1:13;
A11:  dom ((abs f)|Ef`) = dom(abs f) /\Ef` by RELAT_1:61;
A12: dom( f|Ef`) = dom f  /\ Ef` by RELAT_1:61
     .= dom(abs f) /\Ef` by VALUED_1:def 11
     .= dom ((abs f)|Ef`) by RELAT_1:61;
     for x be object st x in dom (f|Ef`) holds f|Ef`.x = (X-->0)|Ef`.x
     proof
     let x be object;
     assume
     A13: x in dom ( f|Ef`);
A14:  x in dom(abs f) & x in Ef` by A13,A12,A11,XBOOLE_0:def 4;
A15: x in  dom ((X-->0)|Ef`) by A10,RELAT_1:62, A14;
A16: (abs f).x = (X-->0)|Ef`.x by A9,A13,A12, FUNCT_1:47
     .= (X-->0).x by A15, FUNCT_1:47
     .= 0 by A13,FUNCOP_1:7;
A17:(Re (f.x))^2 >= 0 & (Im (f.x))^2 >= 0 by XREAL_1:63;
     sqrt((Re (f.x))^2  + (Im (f.x))^2)
     = |.f.x.| by COMPLEX1:def 12
     .=0 by A16,A14,VALUED_1:def 11; then
     (Re (f.x))^2 = 0 & (Im (f.x))^2 = 0 by A17,SQUARE_1:31;
     then
A18: (Re (f.x))^2  + (Im (f.x))^2 = 0;
      f|Ef`.x = f.x by A13,FUNCT_1:47 .= 0 by A18,COMPLEX1:5
      .= (X-->0).x by A13,FUNCOP_1:7
      .=(X-->0)|Ef`.x by A15, FUNCT_1:47;
      hence thesis;
      end;
hence thesis by A8,A9,A12,FUNCT_1:def 11;
end;
