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 Th44:
  for L be Linear_Combination of REAL-NS n,
      S be Linear_Combination of TOP-REAL n
    st L = S
  holds
    sum L = sum S
  proof
    let L be Linear_Combination of REAL-NS n,
        S be Linear_Combination of TOP-REAL n;

    assume
    A1: L = S;

    consider F be FinSequence of REAL-NS n such that
    A2: F is one-to-one & rng F = Carrier L
      & sum L = Sum (L * F) by RLAFFIN1:def 3;

    reconsider E = F as FinSequence of TOP-REAL n by Th4;
    Sum (L * F) = Sum (S * E) by A1;
    hence thesis by A2,A1,RLAFFIN1:def 3;
  end;
