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
  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;
