theorem
  for v1,v2 be Element of S st
      Carrier LS c= {v1,v2} & v1 <> v2 holds sum LS = LS.v1 + LS.v2
  proof
    let v1,v2 be Element of S;
    consider p be FinSequence such that
    A1: rng p={v1,v2} and
    A2: p is one-to-one by FINSEQ_4:58;
    reconsider p as FinSequence of S by A1,FINSEQ_1:def 4;
    assume that
    A3: Carrier LS c={v1,v2} and
    A4: v1<>v2;
    A5: dom LS=the carrier of S by FUNCT_2:def 1;
    A6: Sum<*LS.v1*>=LS.v1 by RVSUM_1:73;
    p=<*v1,v2*> or p=<*v2,v1*> by A1,A2,A4,FINSEQ_3:99;
    then LS*p=<*LS.v1,LS.v2*> or LS*p=<*LS.v2,LS.v1*> by A5,FINSEQ_2:125;
    then Sum(LS*p)=LS.v1+LS.v2 or Sum(LS*p)=LS.v2+LS.v1 by A6,RVSUM_1:74,76;
    hence thesis by A1,A2,A3,Th30;
  end;
