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 Th19:
  g + (h+LG) = (g+h) + LG
  proof
    now let s be Element of G;
      thus(g+(h+LG)).s = (h+LG).(s-g) by Def1
                      .= LG.(s-g-h) by Def1
                      .= LG.(s-(g+h)) by RLVECT_1:27
                      .= ((g+h)+LG).s by Def1;
    end;
    hence thesis;
  end;
