theorem Th31:
  sum ZeroLC S = 0
  proof
    consider F be FinSequence of S such that
    F is one-to-one and
    A1: rng F=Carrier (ZeroLC S) and
    A2: sum ZeroLC S = Sum((ZeroLC S)*F) by Def3;
    F={} by A1,RLVECT_2:def 5;
    hence thesis by A2,RVSUM_1:72;
  end;
