 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 Th39:
  (ex x be VECTOR of Pre-L-CSpace M st f in x & g in x) implies f
  a.e.cpfunc= g,M & f in L1_CFunctions M & g in L1_CFunctions M
proof
  given x be VECTOR of Pre-L-CSpace M such that
A1: f in x and
A2: g in x;
  x in the carrier of Pre-L-CSpace M;
  then x in CCosetSet M by Def19;
  then consider h be PartFunc of X,COMPLEX such that
A3: x=a.e-Ceq-class(h,M) and
  h in L1_CFunctions M;
  ex k be PartFunc of X,COMPLEX st f=k & k in L1_CFunctions M & h in
  L1_CFunctions M & h a.e.cpfunc= k,M by A1,A3;
  then
A4: f a.e.cpfunc= h,M;
  ex j be PartFunc of X,COMPLEX st g=j & j in L1_CFunctions M & h in
  L1_CFunctions M & h a.e.cpfunc= j,M by A2,A3;
  hence thesis by A1,A3,A4,Th24;
end;
