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 Th76:
  for T be RealLinearSpace,
    Lv be Linear_Combination of RLSp2RVSp(T),
    Lr be Linear_Combination of T
    st Lr = Lv
  holds Sum Lr = Sum Lv
  proof
    let T be RealLinearSpace;
    let Lv be Linear_Combination of RLSp2RVSp(T);
    let Lr be Linear_Combination of T;
    assume
    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 RLSp2RVSp(T);
    Carrier Lr = Carrier Lv by A1,Th73;

    hence Sum Lv = Sum (Lv (#) F1) by A2,VECTSP_6:def 6
    .= Sum Lr by A1,A3,Th74;
  end;
