theorem
  r * (s(*)L) = s (*) (r*L)
  proof
    per cases;
    suppose A1: s=0;
      hence r*(s(*)L) = r*ZeroLC(V) by Def2
                         .= r*(0*L) by RLVECT_2:43
                         .= (r*0)*L by RLVECT_2:47
                         .= ZeroLC(V) by RLVECT_2:43
                         .= s(*)(r*L) by A1,Def2;
    end;
    suppose A2: s<>0;
      now let v;
        thus(r*(s(*)L)).v = r*((s(*)L).v) by RLVECT_2:def 11
                         .= r*(L.(s"*v)) by A2,Def2
                         .= (r*L).(s"*v) by RLVECT_2:def 11
                         .= (s(*)(r*L)).v by A2,Def2;
      end;
      hence thesis;
    end;
  end;
