There is plenty of history on the formation of any estate, and Ukraine is no different. In modern times, it is the people that live in a region or country that have the right to decide their future and their organisation.
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Exactly and the people of Donetsk People's Republic and Luhansk People's Republic have decided they want have anything to do with Ukraine.
You nailed it.
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That is quite far from the truth I am afraid. Putin has been supporting a certain group but the local support is nil. Who would like to live under Putin's despotic and inhuman rule?
On Mariupol... it seems that Putin could enter the city. The obvious problem is that there is no longer a city in there. He has simply flattened it and that only leaves ill will and hate. I guess that is not an immediate problem for Putin, yet controlling a territory that is hostile is a costly effort, so this "victory" may become his new Afghanistan, but hey, feel free to consider it a victory if it happens.