 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;

theorem Th8:
  for b2 be Basis of V ex KL be Linear_Combination of V
  st W = Sum KL & Carrier KL c= b2
  proof
    let b2 be Basis of V;
    W in the ModuleStr of V;
    then W in Lin b2 by VECTSP_7:def 3;
    then consider KL1 being Linear_Combination of b2 such that
    A1: W = Sum KL1 by ZMODUL02:64;
    take KL = KL1;
  thus W = Sum KL by A1;
  thus thesis by VECTSP_6:def 4;
end;
