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);

theorem Th51:
f in x implies (abs f) to_power k is_integrable_on M & f in Lp_Functions(M,k)
proof
   assume A1: f in x;
   x in the carrier of Pre-Lp-Space(M,k); then
   x in CosetSet(M,k) by Def11; then
   consider h be PartFunc of X,REAL such that
A2: x=a.e-eq-class_Lp(h,M,k) & h in Lp_Functions(M,k);
   ex g be PartFunc of X,REAL st
    f=g & g in Lp_Functions (M,k) & h a.e.= g,M by A1,A2; then
   ex f0 be PartFunc of X,REAL st
    f=f0 & ex ND be Element of S st M.ND` =0 & dom f0 = ND  &
    f0 is ND-measurable & (abs f0) to_power k is_integrable_on M;
   hence thesis;
end;
