 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);
reserve u,v,w for VECTOR of CLSp_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,A,B for Element of S,
  f,g,h,f1,g1 for PartFunc of X,COMPLEX;
reserve v,u for VECTOR of CLSp_L1Funct M;
reserve v,u for VECTOR of CLSp_AlmostZeroFunct M;

theorem Th32:
  f in L1_CFunctions M & g in L1_CFunctions M implies (a.e-Ceq-class(
  f,M) = a.e-Ceq-class(g,M) iff f a.e.cpfunc= g,M)
proof
  assume that
A1: f in L1_CFunctions M and
A2: g in L1_CFunctions M;
  hereby
    assume a.e-Ceq-class(f,M) = a.e-Ceq-class(g,M);
    then f in {r where r is PartFunc of X,COMPLEX : r in L1_CFunctions M &
    g in L1_CFunctions M & g a.e.cpfunc= r,M } by A1,Th31;
    then ex r be PartFunc of X,COMPLEX st f=r & r in L1_CFunctions M
    & g in L1_CFunctions M & g a.e.cpfunc= r,M;
    hence f a.e.cpfunc= g,M;
  end;
  assume
A3: f a.e.cpfunc= g,M;
  now
    let x be object;
    assume x in a.e-Ceq-class(f,M);
    then consider r be PartFunc of X,COMPLEX such that
A4: x=r & r in L1_CFunctions M and
    f in L1_CFunctions M and
A5: f a.e.cpfunc= r,M;
    g a.e.cpfunc= f,M by A3;
    then g a.e.cpfunc= r,M by A5,Th24;
    hence x in a.e-Ceq-class(g,M) by A2,A4;
  end;
  then
A6: a.e-Ceq-class(f,M) c= a.e-Ceq-class(g,M);
  now
    let x be object;
    assume x in a.e-Ceq-class(g,M);
    then consider r be PartFunc of X,COMPLEX such that
A7: x=r & r in L1_CFunctions M and
    g in L1_CFunctions M and
A8: g a.e.cpfunc= r,M;
    f a.e.cpfunc= r,M by A3,A8,Th24;
    hence x in a.e-Ceq-class(f,M) by A1,A7;
  end;
  then a.e-Ceq-class(g,M) c= a.e-Ceq-class(f,M);
  hence thesis by A6,XBOOLE_0:def 10;
end;
