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;
