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;
