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;
