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 Th72:
for X be non empty set, S be SigmaField of X,
 M be sigma_Measure of S holds addCoset M = addCoset(M,1)
proof
   let X be non empty set;
   let S be SigmaField of X;
   let M be sigma_Measure of S;
A1:CosetSet M = CosetSet(M,1) by Th71;
   now let A,B be Element of CosetSet M;
    A in {a.e-eq-class(f,M) where f is PartFunc of X,REAL
          : f in L1_Functions M}; then
    consider a be PartFunc of X,REAL such that
A2:  A = a.e-eq-class(a,M) & a in L1_Functions M;
    B in {a.e-eq-class(f,M) where f is PartFunc of X,REAL
          : f in L1_Functions M}; then
    consider b be PartFunc of X,REAL such that
A3:  B = a.e-eq-class(b,M) & b in L1_Functions M;
A4: A is Element of CosetSet(M,1) & B is Element of CosetSet(M,1) by Th71;
A5: a in a.e-eq-class(a,M) & b in a.e-eq-class(b,M) by A2,A3,LPSPACE1:38; then
A6: (addCoset M).(A,B) = a.e-eq-class(a+b,M) by A2,A3,LPSPACE1:def 15;
    a+b in L1_Functions M by A2,A3,LPSPACE1:23; then
    ex E be Element of S st M.E`=0 & E = dom(a+b) & (a+b) is E-measurable
       by Lm8; then
    (addCoset M).(A,B) = a.e-eq-class_Lp(a+b,M,1) by A6,Lm12;
    hence (addCoset M).(A,B) = (addCoset(M,1)).(A,B) by A4,A5,A2,A3,Def8;
   end;
   hence addCoset M = addCoset(M,1) by A1,BINOP_1:2;
end;
