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Is it possible to upgrade version while keeping all the message from the last version?
If so, is there a written procedure to do it?
Yes.
Not really.
Get a backup or two of everything (database and phorum directory) before you start.
Normally you upload a newer version of phorum on top of the old one, _after_ saving any customizations.
Your phorum version is old enough that, php, and mysql may have been upgraded by your host.
This may cause upgrade issues.
There are several topics here that mention the issues you may have.
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Most customization's will need to be redone for phorum 5.2.
php 5.5 dropped the deprecated TYPE keyword.
mysql may need sql-mode="" due to defaults changing