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 Th17:
  g + (LG1+LG2) = (g+LG1) + (g+LG2)
  proof
    now let h;
      thus(g+(LG1+LG2)).h = (LG1+LG2).(h-g) by Def1
                         .= LG1.(h-g)+LG2.(h-g) by RLVECT_2:def 10
                         .=(g+LG1).h+LG2.(h-g) by Def1
                         .=(g+LG1).h+(g+LG2).h by Def1
                         .=((g+LG1)+(g+LG2)).h by RLVECT_2:def 10;
    end;
    hence thesis;
  end;
