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 Th53:
(ex f be PartFunc of X,REAL st
   f in Lp_Functions(M,k) & x= a.e-eq-class_Lp(f,M,k)) &
for f be PartFunc of X,REAL st f in x holds
  ex r be Real st
    0<=r & r = Integral(M,(abs f) to_power k) & ||.x.|| =r to_power (1/k)
proof
   x in the carrier of Pre-Lp-Space(M,k); then
   x in CosetSet(M,k) by Def11; then
   ex g be PartFunc of X,REAL st
    x=a.e-eq-class_Lp(g,M,k) & g in Lp_Functions(M,k);
   hence ex f be PartFunc of X,REAL st f in Lp_Functions(M,k) &
    x= a.e-eq-class_Lp(f,M,k);
   consider f be PartFunc of X,REAL such that
A1: f in x & ex r be Real st r = Integral(M,(abs f) to_power k) &
    (Lp-Norm(M,k)).x = r to_power (1/k) by Def12;
   hereby let g be PartFunc of X,REAL;
    assume A2: g in x; then
A3: g in Lp_Functions(M,k) by Th50;
    Integral(M,(abs g) to_power k) = Integral(M,(abs f) to_power k)
      by A1,Th52,A2;
    hence ex r be Real st 0 <= r & r = Integral(M,(abs g) to_power k) &
           ||.x.|| =r to_power (1/k) by A1,A3,Th49;
   end;
end;
