theorem Th35:
  sum (r * L) = r * sum L
  proof
    consider F be FinSequence of V such that
    A1: F is one-to-one and
    A2: rng F=Carrier L and
    A3: sum L=Sum(L*F) by Def3;
    L is Linear_Combination of Carrier L by RLVECT_2:def 6;
    then r*L is Linear_Combination of Carrier L by RLVECT_2:44;
    then A4: Carrier(r*L)c=rng F by A2,RLVECT_2:def 6;
    thus r*sum L = Sum(r*(L*F)) by A3,RVSUM_1:87
                .= Sum((r*L)*F) by Th14
                .= sum(r*L) by A1,A4,Th30;
  end;
