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 Th47:
f in Lp_Functions(M,k) & g in Lp_Functions(M,k) &
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 A1: f in Lp_Functions (M,k) & g in Lp_Functions (M,k) &
    a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(g,M,k); then
A2:(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)
      by Th35;
   f in a.e-eq-class_Lp(g,M,k) by A1,Th38; then
   f a.e.= g,M & a(#)f in Lp_Functions(M,k) &
   a(#)g in Lp_Functions(M,k) by A2,Th37,Th26;
   hence thesis by Th42,LPSPACE1:32;
end;
