reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th5:
  for Lv be Linear_Combination of n-VectSp_over F_Real,
      Lr be Linear_Combination of TOP-REAL n st Lr = Lv
  holds Sum Lr = Sum Lv
proof
  set V=n-VectSp_over F_Real;
  set T=TOP-REAL n;
  let Lv be Linear_Combination of V;
  let Lr be Linear_Combination of T such that
   A1: Lr=Lv;
  consider F be FinSequence of the carrier of T such that
   A2: (F is one-to-one) & rng F=Carrier(Lr) and
   A3: Sum Lr=Sum(Lr(#)F) by RLVECT_2:def 8;
  reconsider F1=F as FinSequence of the carrier of V by Lm1;
  A4: Lr(#)F=Lv(#)F1 by A1,Th3;
  Carrier(Lr)=Carrier(Lv) by A1,Th2;
  hence Sum Lv=Sum(Lv(#)F1) by A2,VECTSP_6:def 6
   .=Sum Lr by A3,A4,Th4;
end;
