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 Th55:
f in x implies x= a.e-eq-class_Lp(f,M,k) & ( ex r be Real st
     0 <=r & r = Integral(M,(abs f) to_power k) & ||.x.|| = r to_power (1/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 g be PartFunc of X,REAL such that
A2: x=a.e-eq-class_Lp(g,M,k) & g in Lp_Functions(M,k);
   g in x by A2,Th38; then
   f a.e.= g,M & f in Lp_Functions(M,k) & g in Lp_Functions(M,k) by A1,Th50;
   hence thesis by Th53,A1,A2,Th42;
end;
