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 Th73:
for X be non empty set, S be SigmaField of X,
 M be sigma_Measure of S holds zeroCoset M = zeroCoset(M,1)
proof
   let X be non empty set;
   let S be SigmaField of X;
   let M be sigma_Measure of S;
   reconsider z = zeroCoset(M,1) as Element of CosetSet M by Th71;
   X-->0 in Lp_Functions(M,1) by Th23; then
   ex E be Element of S st
    M.E`=0 & dom (X-->0) = E & (X-->0) is E-measurable by Th35; then
A1:z = a.e-eq-class(X-->0,M) by Lm12;
   X-->0 in L1_Functions M by Th56;
   hence thesis by A1,LPSPACE1:def 16;
end;
