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 Th71:
for X be non empty set, S be SigmaField of X,
 M be sigma_Measure of S holds CosetSet M = CosetSet(M,1)
proof
   let X be non empty set;
   let S be SigmaField of X;
   let M be sigma_Measure of S;
   now let x be object;
    assume x in CosetSet M; then
    consider g be PartFunc of X,REAL such that
A1:  x = a.e-eq-class(g,M) & g in L1_Functions M;
A2: g is_integrable_on M &
    ex E be Element of S st M.E`=0 & E = dom g & g is E-measurable
       by A1,Lm8; then
A3: x = a.e-eq-class_Lp(g,M,1) by A1,Lm12;
    (abs g) to_power 1 = abs g by Th8; then
    (abs g) to_power 1 is_integrable_on M by A2,MESFUNC6:94; then
    g in Lp_Functions(M,1) by A2;
    hence x in CosetSet(M,1) by A3;
   end; then
A4:CosetSet M c= CosetSet(M,1);
   now let x be object;
    assume x in CosetSet(M,1); then
    consider g be PartFunc of X,REAL such that
A5:  x = a.e-eq-class_Lp(g,M,1) & g in Lp_Functions(M,1);
    consider E be Element of S such that
A6:  M.E` = 0 & dom g = E & g is E-measurable by A5,Th35;
A7: x = a.e-eq-class(g,M) by A5,A6,Lm12;
    reconsider D = E` as Element of S by MEASURE1:34;
A8: M.D = 0 & dom g = D` by A6;
    (abs g) to_power 1 is_integrable_on M by A5,Lm9; then
    abs g is_integrable_on M by Th8; then
    g is_integrable_on M by A6,MESFUNC6:94; then
    g in L1_Functions M by A8;
    hence x in CosetSet M by A7;
   end; then
   CosetSet(M,1) c= CosetSet M;
   hence thesis by A4;
end;
