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 Th27:
  r(*)(s(*)LR)=(r*s)(*)LR
  proof
    per cases;
    suppose A1: r=0 or s=0;
      then (r*s)(*)LR=ZeroLC(R) by Def2;
      hence thesis by A1,Def2,Th26;
    end;
    suppose A2: r<>0 & s<>0;
      now let v be Element of R;
        thus(r(*)(s(*)LR)).v = (s(*)LR).(r"*v) by A2,Def2
                            .= LR.(s"*(r"*v)) by A2,Def2
                            .= LR.((s"*r")*v) by RLVECT_1:def 7
                            .= LR.((r*s)"*v) by XCMPLX_1:204
                            .= ((r*s)(*)LR).v by A2,Def2;
      end;
      hence thesis;
    end;
  end;
