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 Th23:
  for Lr be Linear_Combination of REAL-NS n,
      Lt be Linear_Combination of TOP-REAL n st Lr = Lt
  holds Sum Lr = Sum Lt
  proof
    let Lr be Linear_Combination of REAL-NS n,
        Lt be Linear_Combination of TOP-REAL n;
    assume
    A1: Lr = Lt;
    set R = REAL-NS n;
    set T = TOP-REAL n;
    consider Ft be FinSequence of the carrier of (TOP-REAL n) such that
    A2: (Ft is one-to-one & rng Ft = Carrier Lt) and
    A3: Sum Lt = Sum (Lt (#) Ft) by RLVECT_2:def 8;
    reconsider Fr = Ft as FinSequence of the carrier of (REAL-NS n) by Th4;
    thus Sum Lt
     = Sum (Lr (#) Fr) by A1,A3,Th19,Th21
    .= Sum Lr by A1,A2,RLVECT_2:def 8;
  end;
