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 Th52:
f in x & g in x implies f a.e.= g,M &
  Integral(M,(abs f) to_power k) = Integral(M,(abs g) to_power k)
proof
   assume f in x & g in x; then
   f a.e.= g,M & f in Lp_Functions(M,k) & g in Lp_Functions(M,k) by Th50;
   hence thesis by Th48;
end;
