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 Th39:
(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(f,M,k) = a.e-eq-class_Lp(g,M,k) implies
  f a.e.= g,M
proof
   assume that
A1:(ex E be Element of S st M.(E`)=0 & E = dom g & g is E-measurable) and
A2: a.e-eq-class_Lp(f,M,k) <> {} and
A3: a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(g,M,k);
   consider x be object such that
A4: x in a.e-eq-class_Lp(f,M,k) by A2,XBOOLE_0:def 1;
   consider r be PartFunc of X,REAL such that
A5: x = r & r in Lp_Functions(M,k) & f a.e.= r,M by A4;
   r a.e.= g,M by A1,A3,A4,A5,Th37;
   hence thesis by A5,LPSPACE1:30;
end;
