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;
reserve v,u for VECTOR of RLSp_L1Funct M;
reserve v,u for VECTOR of RLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-Space M;
reserve x,y for Point of L-1-Space M;

theorem Th53:
  f in L1_Functions M & Integral(M,abs f) = 0 implies f a.e.= X--> 0,M
proof
  assume that
A1: f in L1_Functions M and
A2: Integral(M,abs f) = 0;
  set g = X-->0;
  defpred P[Element of NAT,set] means $2=great_eq_dom(abs f,1/($1+1)) & M.($2)
  =0;
  consider f1 be PartFunc of X,REAL such that
A3: f=f1 and
A4: ex ND be Element of S st M.ND=0 & dom f1 = ND` & f1 is_integrable_on
  M by A1;
A5: abs f is_integrable_on M by A3,A4,Th44;
  then R_EAL abs f is_integrable_on M;
  then consider E be Element of S such that
A6: E = dom R_EAL abs f and
A7: R_EAL abs f is E-measurable;
A8: abs f is E-measurable by A7;
  now
    let x be object;
    assume x in dom abs f;
    then (abs f).x = |.f.x qua Complex.| by VALUED_1:def 11;
    hence 0 <= (abs f).x by COMPLEX1:46;
  end;
  then
A9: abs f is nonnegative by MESFUNC6:52;
A10: now
    let n be Element of NAT;
    reconsider r=1/(n+1) as Element of REAL by XREAL_0:def 1;
    reconsider Br=E /\ great_eq_dom(abs f,r) as Element of S by A6,A8,
MESFUNC6:13;
    set g = Br --> r;
A11: dom g = Br by FUNCT_2:def 1;
A12: for x be set st x in dom g holds g.x = r by FUNCOP_1:7;
    reconsider g as PartFunc of X,REAL by RELSET_1:7;
A13: (abs f)|Br is_integrable_on M by A5,MESFUNC6:91;
    for x be object st x in dom g holds 0 <= g.x by A12;
    then g is nonnegative by MESFUNC6:52;
    then
A14: 0 <= Integral(M,g) by A11,A12,Th52,MESFUNC6:84;
A15: dom abs g =dom g by VALUED_1:def 11;
A16: now
      let x be Element of X;
      assume
A17:  x in dom abs g;
      then (abs g).x = |.g.x qua Complex.| by VALUED_1:def 11;
      then (abs g).x = |.r qua Complex.| by A12,A15,A17;
      then (abs g).x = r by ABSVALUE:def 1;
      hence (abs g).x = g.x by A12,A15,A17;
    end;
A18: dom ((abs f)|Br) = dom (abs f) /\ Br by RELAT_1:61
      .= Br by A6,XBOOLE_1:17,28;
    then
A19: dom g = dom ((abs f)|Br) by FUNCT_2:def 1;
A20: now
      let x be Element of X;
      assume
A21:  x in dom g;
      then x in E /\ great_eq_dom(abs f,r) by FUNCT_2:def 1;
      then x in great_eq_dom(abs f,r) by XBOOLE_0:def 4;
      then
A22:  ex y being Real st y=(abs f).x & r <= y by MESFUNC6:6;
      |.g.x qua Complex.| = |.r qua Complex.| by A12,A21;
      then |.g.x qua Complex.| = r by ABSVALUE:def 1;
      hence |.g.x qua Complex.| <= ((abs f)|Br).x by A19,A21,A22,FUNCT_1:47;
    end;
    g is Br-measurable by A11,A12,Th52;
    then Integral(M,abs g) <= Integral(M,(abs f)|Br) by A11,A18,A13,A20,
MESFUNC6:96;
    then
A23: Integral(M, g) <= Integral(M,(abs f)|Br) by A15,A16,PARTFUN1:5;
A24: for x be object st x in dom g holds g.x = r by A11,FUNCOP_1:7;
    reconsider rr=r as R_eal by XXREAL_0:def 1;
AAA: 1/(n+1) = 1 qua ExtReal/(n+1);
    Integral(M,(abs f)|Br) <= Integral(M,(abs f)|E)
    by A6,A8,A9,MESFUNC6:87,XBOOLE_1:17;
    then Integral(M,g) = 0 by A2,A6,A23,A14,RELAT_1:68;
    then rr * M.Br = 0 by A11,A24,MESFUNC6:97;
    then
A25: M.Br = 0 by AAA;
    for x be object st x in great_eq_dom(abs f,r) holds x in dom abs f
    by MESFUNC6:6;
    then great_eq_dom(abs f,r) c= E by A6;
    hence ex B be Element of S st P[n,B] by A25,XBOOLE_1:28;
  end;
  consider F be sequence of S such that
A26: for n be Element of NAT holds P[n,F.n] from FUNCT_2:sch 3(A10);
  now
    let y be object;
    assume y in union rng F;
    then consider B be set such that
A27: y in B and
A28: B in rng F by TARSKI:def 4;
    consider n be object such that
A29: n in NAT and
A30: B=F.n by A28,FUNCT_2:11;
    reconsider m=n as Element of NAT by A29;
A31: y in great_eq_dom(abs f,1/(m+1)) by A26,A27,A30;
    then
A32: y in dom abs f by MESFUNC6:6;
    then
A33: y in dom f by VALUED_1:def 11;
A34: (abs f).y = |.f.y qua Complex.| by A32,VALUED_1:def 11;
AAA: 1/(m+1) = 1 qua ExtReal/(m+1);
    1/(m+1) <= (abs f).y by A31,MESFUNC1:def 14;
    then (abs f).y <> 0 by AAA;
    then f.y <> 0 by A34,ABSVALUE:2;
    hence y in {x where x is Element of X : x in dom f & f.x <> 0} by A33;
  end;
  then
A35: union rng F c= {x where x is Element of X : x in dom f & f.x <> 0};
  consider ND be Element of S such that
A36: M.ND=0 and
A37: dom f1 = ND` and
  f1 is_integrable_on M by A4;
  reconsider EQ = union rng F \/ ND as Element of S;
A38: EQ` = ND` /\ (union rng F)` by XBOOLE_1:53;
  then
A39: EQ` c= dom f by A3,A37,XBOOLE_1:17;
  dom g = X by FUNCOP_1:13;
  then
A40: dom(g|EQ`) = EQ` by RELAT_1:62;
A41: dom(f|EQ`) = EQ` by A3,A37,A38,RELAT_1:62,XBOOLE_1:17;
  now
    let y be object;
    assume y in {x where x is Element of X : x in dom f & f.x <> 0};
    then consider z be Element of X such that
A42: y=z and
A43: z in dom f and
A44: f.z <> 0;
A45: z in dom abs f by A43,VALUED_1:def 11;
    then (abs f).z = |.f.z qua Complex.| by VALUED_1:def 11;
    then 0 < (abs f).z by A44,COMPLEX1:47;
    then consider n be Element of NAT such that
A46: 1/(n+1) < (abs f).z - 0 by MESFUNC1:10;
    z in great_eq_dom(abs f,1/(n+1)) by A45,A46,MESFUNC6:6;
    then
A47: y in F.n by A26,A42;
    F.n in rng F by FUNCT_2:4;
    hence y in union rng F by A47,TARSKI:def 4;
  end;
  then {x where x is Element of X : x in dom f & f.x <> 0} c= union rng F;
  then
A48: {x where x is Element of X : x in dom f & f.x <> 0 } = union rng F by A35,
XBOOLE_0:def 10;
A49: now
    let x be set;
    assume
A50: x in EQ`;
    then x in (union rng F)` by A38,XBOOLE_0:def 4;
    then not x in union rng F by XBOOLE_0:def 5;
    hence f.x = 0 by A48,A39,A50;
  end;
  now
    let y be object;
    assume
A51: y in dom(f|EQ`);
    then (f|EQ`).y = f.y by FUNCT_1:47;
    then (f|EQ`).y =0 by A41,A49,A51;
    then (f|EQ`).y =g.y by A51,FUNCOP_1:7;
    hence (f|EQ`).y =(g|EQ`).y by A41,A40,A51,FUNCT_1:47;
  end;
  then
A52: f|EQ` = g|EQ` by A39,A40,FUNCT_1:2,RELAT_1:62;
  now
    let A be set;
    assume A in rng F;
    then consider n be object such that
A53: n in NAT and
A54: A=F.n by FUNCT_2:11;
    reconsider n as Element of NAT by A53;
    M.(F.n) =0 by A26;
    hence A is measure_zero of M by A54,MEASURE1:def 7;
  end;
  then
A55: union rng F is measure_zero of M by MEASURE2:14;
  ND is measure_zero of M by A36,MEASURE1:def 7;
  then EQ is measure_zero of M by A55,MEASURE1:37;
  then M.EQ=0 by MEASURE1:def 7;
  hence thesis by A52;
end;
