theorem Th42:
  f in L1_Functions M & g in L1_Functions M & a.e-eq-class(f,M) =
  a.e-eq-class(g,M) implies a.e-eq-class(a(#)f,M) = a.e-eq-class(a(#)g,M)
proof
  assume that
A1: f in L1_Functions M & g in L1_Functions M and
A2: a.e-eq-class(f,M) = a.e-eq-class(g,M);
  f a.e.= g,M by A1,A2,Th39;
  then
A3: a(#)f a.e.= a(#)g,M by Th32;
  a(#)f in L1_Functions M & a(#)g in L1_Functions M by A1,Th24;
  hence thesis by A3,Th39;
end;
