reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;
reserve v,u for VECTOR of RLSp_AlmostZeroFunct M;

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;
