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 Th39:
  f in L1_Functions M & g in L1_Functions M implies (a.e-eq-class(
  f,M) = a.e-eq-class(g,M) iff f a.e.= g,M)
proof
  assume that
A1: f in L1_Functions M and
A2: g in L1_Functions M;
  hereby
    assume a.e-eq-class(f,M) = a.e-eq-class(g,M);
    then f in {r where r is PartFunc of X,REAL : r in L1_Functions M & g in
    L1_Functions M & g a.e.= r,M } by A1,Th38;
    then ex r be PartFunc of X,REAL st f=r & r in L1_Functions M & g in
    L1_Functions M & g a.e.= r,M;
    hence f a.e.= g,M;
  end;
  assume
A3: f a.e.= g,M;
  now
    let x be object;
    assume x in a.e-eq-class(f,M);
    then consider r be PartFunc of X,REAL such that
A4: x=r & r in L1_Functions M and
    f in L1_Functions M and
A5: f a.e.= r,M;
    g a.e.= f,M by A3;
    then g a.e.= r,M by A5,Th30;
    hence x in a.e-eq-class(g,M) by A2,A4;
  end;
  then
A6: a.e-eq-class(f,M) c= a.e-eq-class(g,M);
  now
    let x be object;
    assume x in a.e-eq-class(g,M);
    then consider r be PartFunc of X,REAL such that
A7: x=r & r in L1_Functions M and
    g in L1_Functions M and
A8: g a.e.= r,M;
    f a.e.= r,M by A3,A8,Th30;
    hence x in a.e-eq-class(f,M) by A1,A7;
  end;
  then a.e-eq-class(g,M) c= a.e-eq-class(f,M);
  hence thesis by A6,XBOOLE_0:def 10;
end;
