We elucidate why the anomalous high strength of double network gels is obtained when the second network is polymerized without any cross-linkers. We have synthesized truly independent-DN gels (named "t-DN" gels), which do not have any covalent bonds between the first and the second networks, and showed that the t-DN gels cannot be toughened by the un-cross-linked second network. It means that the high strength of Usual DIN gels without the cross-linker of the second network is actually achieved by the interconnection between the two networks through covalent bonds (so usual DN gels were named "c-DN" gels). Further, we found that the t-DN gels become stronger than the c-DN gels when the second network is loosely cross-linked. As the t-DN gels have a more simple structure than the c-DN gels, we expect that the toughening mechanism of DN gels will be clarified by studying the t-DN gels in the future.