This may help picture the situation in series. Trace the route to umn.edu from
my home desktop:

iznogoud at bigpapa:~> traceroute umn.edu
traceroute to umn.edu (134.84.119.107), 30 hops max, 60 byte packets
 1  192.168.0.1 (192.168.0.1)  10.194 ms  11.361 ms  11.557 ms
 2  192.1.0.1 (192.1.0.1)  12.180 ms  12.380 ms  13.147 ms
 3  stpl-dsl-gw11.stpl.qwest.net (207.109.2.11)  27.331 ms  27.768 ms  31.007 ms
 4  stpl-agw1.inet.qwest.net (207.109.3.81)  31.311 ms  31.733 ms  35.064 ms
...

The first 192.168.0.1 is the D-Link WAN. The second is the 192.1.0.1, which is
the modem+router LAN, and this is also the gateway address of the D-Link's WAN.
The D-Link's WAN has an IP like 192.1.0.2 or something like that.

Say the WAN of the DSL-model+router dies. The D-Link will have the 192.0.1.*
link dead, but the 192.168.0.* network will work just fine. All your computers
will live in that network and will work fine.