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);

theorem
f in Lp_Functions(M,k) &
(ex E be Element of S st M.(E`)=0 & E = dom g & g is E-measurable) &
a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(g,M,k)
  implies f a.e.= g,M
proof
   assume that
A1: f in Lp_Functions(M,k) and
A2: (ex E be Element of S st M.(E`)=0 & E = dom g & g is E-measurable) and
A3: a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(g,M,k);
   a.e-eq-class_Lp(f,M,k) is non empty by A1,Th38;
   hence thesis by A2,A3,Th39;
end;
