reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;

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;
