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;
