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 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;
