theorem
  for Lv be Linear_Combination of n-VectSp_over F_Real,
      Lr be Linear_Combination of REAL-NS n st Lr = Lv
  holds Sum Lr = Sum 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 Lt = Lr as Linear_Combination of TOP-REAL n by Th11;
    Sum Lt = Sum Lr by Th23;
    hence thesis by A1,MATRTOP2:5;
  end;
