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 Th35:
  sum (r * L) = r * sum L
  proof
    consider F be FinSequence of V such that
    A1: F is one-to-one and
    A2: rng F=Carrier L and
    A3: sum L=Sum(L*F) by Def3;
    L is Linear_Combination of Carrier L by RLVECT_2:def 6;
    then r*L is Linear_Combination of Carrier L by RLVECT_2:44;
    then A4: Carrier(r*L)c=rng F by A2,RLVECT_2:def 6;
    thus r*sum L = Sum(r*(L*F)) by A3,RVSUM_1:87
                .= Sum((r*L)*F) by Th14
                .= sum(r*L) by A1,A4,Th30;
  end;
