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 Th58:
Integral(M,(abs(X-->0)) to_power k) = 0
proof
A1:for x be object st x in dom (X-->0) holds 0 <= (X-->0).x;
   then Integral(M,(abs(X-->0)) to_power k)
     = Integral(M,(X-->0) to_power k) by Th14,MESFUNC6:52
    .= Integral(M,(X-->0)) by Th12
    .= Integral(M,abs(X-->0)) by A1,Th14,MESFUNC6:52;
   hence thesis by LPSPACE1:54;
end;
