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 Th56:
X --> 0 in L1_Functions M
proof
   reconsider ND = {} as Element of S by MEASURE1:34;
A1:M.ND =0 by VALUED_0:def 19;
   X --> In(0,REAL) is Function of X,REAL by FUNCOP_1:46; then
A2:dom (X --> 0) = ND` by FUNCT_2:def 1;
   for x be Element of X st x in dom (X --> 0) holds (X --> 0).x = 0
     by FUNCOP_1:7; then
   X --> 0 is_integrable_on M by A2,Th15;
   hence thesis by A1,A2;
end;
