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;

theorem Th32:
  f a.e.= g,M implies a(#)f a.e.= a(#)g,M
proof
  assume f a.e.= g,M;
  then consider E being Element of S such that
A1: M.E = 0 and
A2: f|E`=g|E`;
  (a(#)f)|E` = a(#)(g|E`) by A2,RFUNCT_1:49
    .= (a(#)g)|E` by RFUNCT_1:49;
  hence thesis by A1;
end;
