Uninstalling VS 2013 when VS 2015 is present

I don't know why, but recently VS 2015 started to do weird stuff, refusing to open old projects, etc. I tried repairing it, but same problems. So, I'm trying to uninstall it, but it stops around 75%. I thought the problem might come from my "old" VS 2013, and a little bit of disk space is always welcome, so I'm trying to uninstall VS 2013 now. It stops around 10%, with this error (translated from French):

Installation failed: Couldn't pursue the installation, because some required components have failed.

Last lines from the log file:

[3E7C:09EC][2016-06-17T15:52:42]i000: MUX:  Installation size in bytes for package: vs_professionalcoreres MaxAppDrive: 0  MaxSysDrive: 269455360  AppDrive: 0  SysDrive: 186781696
[3E7C:09EC][2016-06-17T15:52:42]i000: MUX:  Return Code:0x0 Msi Messages:0 Result Detail:RollbackCleanup Restart:None
[3E7C:09EC][2016-06-17T15:52:42]i000: MUX:  Reset execution Result
[3E7C:09EC][2016-06-17T15:52:42]i000: MUX:  Reset Result
[3E7C:09EC][2016-06-17T15:52:42]i319: Applied rollback package: vs_professionalcoreres, result: 0x0, restart: None
[2020:233C][2016-06-17T15:52:42]i325: Registering dependency: {bcb1dbb2-9952-4189-a656-cf0820910916} on package provider: Microsoft.VS.professionalcoreres_fra,v12, package: vs_professionalcoreres
[2020:233C][2016-06-17T15:52:42]i372: Session end, registration key: SOFTWARE\Microsoft\Windows\CurrentVersion\Uninstall\{bcb1dbb2-9952-4189-a656-cf0820910916}, resume: ARP, restart: None, disable resume: No
[2020:233C][2016-06-17T15:52:42]i371: Updating session, registration key: SOFTWARE\Microsoft\Windows\CurrentVersion\Uninstall\{bcb1dbb2-9952-4189-a656-cf0820910916}, resume: ARP, restart initiated: No, disable resume: No
[3E7C:09EC][2016-06-17T15:52:42]i000: MUX:  Apply Complete: Disk Space Used in bytes for Installation:  MaxAppDrive: 0  MaxSysDrive: 86581248  AppDrive: 0  SysDrive: 3907584
[3E7C:09EC][2016-06-17T15:52:42]i000: MUX:  Free Disk Space after install:  SystemDrive C:\ 103698874368 bytes  AppDrive C:\ 103698874368 bytes
[3E7C:09EC][2016-06-17T15:52:42]i000: MUX:  Go to Finished page.
[3E7C:09EC][2016-06-17T15:52:42]i399: Apply complete, result: 0x80070643, restart: None, ba requested restart:  No