Hi Ziad,
In reality, when R3 receives an update from R1 or R2 about the destination behind R4, it won’t install them since these don’t pass the feasibility condition (AD of the feasible successor has to be lower than FD of successor).
R3 will only advertise its successor route to other neighbors. In this topology, R4 is the successor route so normally R3 would advertise this route to R4. However, because of split horizon (don’t advertise a route to a neighbor if you learned that route from the neighbor)…this route is not advertised.
It might be helpful to see all of this in action. Connect four routers like in the topology picture, then load these four configs:
R1-show-run-2018-01-17-15-58-39-clean.txt (270 Bytes)
R2-show-run-2018-01-17-15-58-43-clean.txt (270 Bytes)
R3-show-run-2018-01-17-15-58-46-clean.txt (366 Bytes)
R4-show-run-2018-01-17-15-58-50-clean.txt (261 Bytes)
If you want to see the EIGRP updates, you can start Wireshark and use this filter:
eigrp.opcode == 1
It will show you the route that is advertised and the metric.
Seeing this in action probably makes it easier to understand.
Rene