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
  v + (r*L) = r * (v+L)
  proof
    now let w;
      thus(v+(r*L)).w = (r*L).(w-v) by Def1
                     .= r*(L.(w-v)) by RLVECT_2:def 11
                     .= r*((v+L).w) by Def1
                     .= (r*(v+L)).w by RLVECT_2:def 11;
    end;
    hence thesis;
  end;
