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
(ex E be Element of S st M.(E`) = 0 & dom f = E & f is E-measurable) &
(ex E be Element of S st M.(E`) = 0 & dom g = E & g is E-measurable) &
a.e-eq-class_Lp(f,M,k) is non empty &
a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(g,M,k) implies
  a.e-eq-class_Lp(a(#)f,M,k) = a.e-eq-class_Lp(a(#)g,M,k)
proof
   assume
 (ex E be Element of S st M.(E`) = 0 & dom f = E & f is E-measurable) &
    (ex E be Element of S st M.(E`) = 0 & dom g = E & g is E-measurable) &
    a.e-eq-class_Lp(f,M,k) is non empty &
    a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(g,M,k);
   then a(#)f a.e.= a(#)g,M by Th39,LPSPACE1:32;
   hence thesis by Th41;
end;
