reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);
reserve x for Point of Pre-Lp-Space(M,k);
reserve x,y for Point of Lp-Space(M,k);

theorem Th57:
f in Lp_Functions(M,k) & Integral(M,(abs f) to_power k) = 0 implies
   f a.e.= X-->0,M
proof
   assume that
A1: f in Lp_Functions(M,k) and
A2: Integral(M,(abs f) to_power k) = 0;
   ex h be PartFunc of X,REAL st
    f=h & ex ND be Element of S st M.ND` =0 & dom h = ND &
    h is ND-measurable & (abs h) to_power k is_integrable_on M by A1; then
   consider NDf be Element of S such that
A3: M.NDf`=0 & dom f = NDf & f is NDf-measurable &
    (abs f) to_power k is_integrable_on M;
   reconsider t = (abs f) to_power k as PartFunc of X,REAL;
   reconsider ND = NDf` as Element of S by MEASURE1:34;
A4:dom t = dom (abs f) by MESFUN6C:def 4; then
A5:dom t = NDf by A3,VALUED_1:def 11;
   dom t = ND` by A4,A3,VALUED_1:def 11; then
A6:t in L1_Functions M by A3;
   abs t = t by Th14; then
   t a.e.= X-->0,M by A2,A6,LPSPACE1:53; then
   consider ND1 be Element of S such that
A7: M.ND1 = 0 & (abs f) to_power k|ND1` = (X-->0)|ND1`;
   set ND2= ND \/ ND1;
   ND is measure_zero of M & ND1 is measure_zero of M
     by A3,A7,MEASURE1:def 7; then
   ND2 is measure_zero of M by MEASURE1:37; then
A8:M.ND2 = 0 by MEASURE1:def 7;
A9:ND2` c= ND` & ND2` c= ND1` by XBOOLE_1:7,34;
   dom(X-->0) = X by FUNCOP_1:13; then
A10:dom((X-->0)|ND2`) = ND2` by RELAT_1:62;
A11:dom(f|ND2`) = ND2` by A3,A9,RELAT_1:62;
   for x be object st x in dom (f|ND2`) holds (f|ND2`).x = ((X-->0)|ND2`).x
   proof
    let x be object;
    assume A12: x in dom (f|ND2`);
A13: now assume f.x <> 0; then
     |.f.x.| > 0 by COMPLEX1:47; then
     |.f.x qua Complex.| to_power k <> 0 by POWER:34; then
     ((abs f).x) to_power k <> 0 by VALUED_1:18; then
A14:  ((abs f) to_power k).x <> 0 by A5,A9,A12,A11,MESFUN6C:def 4;
     ((X-->0)|ND1`).x = (X-->0).x by A9,A12,A11,FUNCT_1:49; then
     ((X-->0)|ND1`).x = 0 by A12,FUNCOP_1:7;
     hence contradiction by A14,A7,A9,A12,A11,FUNCT_1:49;
    end;
    ((X-->0)|ND2`).x =(X-->0).x by A11,A12,FUNCT_1:49; then
    ((X-->0)|ND2`).x = 0 by A12,FUNCOP_1:7;
    hence thesis by A11,A12,A13,FUNCT_1:49;
   end; then
   f |ND2` = (X-->0)|ND2` by A10,A11,FUNCT_1:def 11;
   hence thesis by A8;
end;
