 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;

theorem Th21:
  f=u implies u+(-1r)*u = (X --> 0c)|dom f &
  ex v,g be PartFunc of X,COMPLEX st v in L1_CFunctions M
  & g in L1_CFunctions M & v = u+(-1r)*u & g = X --> 0c & v a.e.cpfunc= g,M
proof
  reconsider u2=u as VECTOR of CLSp_PFunctX by TARSKI:def 3;
  reconsider h = u2+(-1r)*u2 as Element of PFuncs(X,COMPLEX);
  set g = X-->0c;
  u+(-1r)*u in L1_CFunctions M;
  then consider v be PartFunc of X,COMPLEX such that
A1: v=u+(-1r)*u and
  ex ND be Element of S st M.ND=0 & dom v = ND` & v is_integrable_on M;
  assume
A2: f=u;
    reconsider u9=u2 as Element of PFuncs(X,COMPLEX);
A3: h = (addcpfunc X).(u2,(multcomplexcpfunc X).(-1r,u2))
     .= (CPFuncZero X)|(dom f) by A2,Th10;
  u in L1_CFunctions M; then
  ex uu1 be PartFunc of X,COMPLEX 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;
  [R,u] in [:COMPLEX,L1_CFunctions M:]; then
A6: (-1r)*u2=(-1r)*u by FUNCT_1:49;
  hence u+(-1r)*u = (X --> 0)|dom f by A3,ZFMISC_1:87,FUNCT_1:49;
  v|ND` = g|ND`|ND` by A3,A6,A1,A5,ZFMISC_1:87,FUNCT_1:49;
  then
A7: v|ND` = g|ND` by FUNCT_1:51;
  g in L1_CFunctions M by Lm3;
  hence thesis by A1,A4,A7,Def11;
end;
