theorem Th32:
  for v be Element of S st Carrier LS c= {v} holds sum LS = LS.v
  proof
    let v be Element of S;
    consider p be FinSequence such that
    A1: rng p={v} and
    A2: p is one-to-one by FINSEQ_4:58;
    reconsider p as FinSequence of S by A1,FINSEQ_1:def 4;
    dom LS=the carrier of S & p=<*v*> by A1,A2,FINSEQ_3:97,FUNCT_2:def 1;
    then A3: LS*p=<*LS.v*> by FINSEQ_2:34;
    assume Carrier LS c={v};
    hence sum LS = Sum(LS*p) by A1,A2,Th30
                .= LS.v by A3,RVSUM_1:73;
  end;
