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
  for Lv be Linear_Combination of n-VectSp_over F_Real,
      Lr be Linear_Combination of REAL-NS n
    st Lr = Lv
  holds Carrier Lr = Carrier Lv
  proof
    let Lv be Linear_Combination of n-VectSp_over F_Real,
        Lr be Linear_Combination of REAL-NS n;
    assume
    A1: Lr = Lv;
    reconsider Lr1 = Lr as Linear_Combination of TOP-REAL n by Th11;
    Carrier Lr1 = Carrier Lv by A1, MATRTOP2:2;
    hence thesis;
  end;
