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 Th37:
  f in L1_Functions M & g in L1_Functions M implies (g a.e.= f,M
  iff g in a.e-eq-class(f,M))
proof
  assume
A1: f in L1_Functions M & g in L1_Functions M;
  hereby
    assume g a.e.= f,M;
    then f a.e.= g,M;
    hence g in a.e-eq-class(f,M) by A1;
  end;
  hereby
    assume g in a.e-eq-class(f,M);
    then ex r be PartFunc of X,REAL st g=r & r in L1_Functions M & f in
    L1_Functions M & f a.e.= r,M;
    hence g a.e.= f,M;
  end;
end;
