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 Th46:
  (ex x be VECTOR of Pre-L-Space M st f in x & g in x) implies f
  a.e.= g,M & f in L1_Functions M & g in L1_Functions M
proof
  assume ex x be VECTOR of Pre-L-Space M st f in x & g in x;
  then consider x be VECTOR of Pre-L-Space M such that
A1: f in x and
A2: g in x;
  x in the carrier of Pre-L-Space M;
  then x in CosetSet M by Def18;
  then consider h be PartFunc of X,REAL such that
A3: x=a.e-eq-class(h,M) and
  h in L1_Functions M;
  ex k be PartFunc of X,REAL st f=k & k in L1_Functions M & h in
  L1_Functions M & h a.e.= k,M by A1,A3;
  then
A4: f a.e.= h,M;
  ex j be PartFunc of X,REAL st g=j & j in L1_Functions M & h in
  L1_Functions M & h a.e.= j,M by A2,A3;
  hence thesis by A1,A3,A4,Th30;
end;
