reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  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 by RLVECT_1:1;
  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 VECTSP_7:7;
  take KL = KL1;
  thus W = Sum KL by A1;
  thus thesis by VECTSP_6:def 4;
end;
