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 Th29:
  for F be FinSequence of S st Carrier LS misses rng F holds Sum (LS*F) = 0
  proof
    let F be FinSequence of S such that
    A1: Carrier LS misses rng F;
    set LF=LS*F;
    set LF0=len LF|->(0 qua Real);
    A2: now let k be Nat;
            assume A3: 1<=k & k<=len LF;
            A4: k in dom LF by A3,FINSEQ_3:25;
            then k in dom F by FUNCT_1:11;
            then F.k in rng F by FUNCT_1:def 3;
            then A5: dom LS=the carrier of S & not F.k in Carrier LS
              by A1,FUNCT_2:def 1,XBOOLE_0:3;
            LF.k=LS.(F.k) & F.k in dom LS by A4,FUNCT_1:11,12;
            hence LF.k=LF0.k by A5;
          end;
    len LF0=len LF by CARD_1:def 7;
    then LF=LF0 by A2;
    hence thesis by RVSUM_1:81;
  end;
