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;

theorem Th27:
  f=u implies u+(-1)*u = (X --> 0)|dom f & ex v,g be PartFunc of X
,REAL st v in L1_Functions M & g in L1_Functions M & v = u+(-1)*u & g = X --> 0
  & v a.e.= g,M
proof
  reconsider u2=u as VECTOR of RLSp_PFunctX by TARSKI:def 3;
  reconsider h = u2+(-1)*u2 as Element of PFuncs(X,REAL);
  set g = X-->0;
  u+(-1)*u in L1_Functions M;
  then consider v be PartFunc of X,REAL such that
A1: v=u+(-1)*u and
  ex ND be Element of S st M.ND=0 & dom v = ND` & v is_integrable_on M;
  assume
A2: f=u;
  then
A3: h = (RealPFuncZero X)|dom f by Th16;
  u in L1_Functions M;
  then
  ex uu1 be PartFunc of X,REAL st uu1=u & ex ND be Element of S st M.ND=0
  & dom uu1 = ND` & uu1 is_integrable_on M;
  then consider ND be Element of S such that
A4: M.ND=0 and
A5: dom f = ND` and
  f is_integrable_on M by A2;
A6: (-1)*u2=(-1)*u by Th5;
  hence u+(-1)*u = (X --> 0)|dom f by A3,Th4;
  v|ND` = g|ND`|ND` by A3,A6,A1,A5,Th4;
  then v|ND` = g|ND` by FUNCT_1:51;
  then
A7: v a.e.= g,M by A4;
  g in L1_Functions M by Lm3;
  hence thesis by A1,A7;
end;
