This error may happen when you try to update from 10166 to 10240 (or the RTM version). 

TH1 Professional 10240 - Error 0x80200013
How to resolve TH1 Professional 10240 – Error 0x80200013?

The error is “TH1 Professional 10240 – Error 0x80200013” and when you open the ReportingEvents.log file, you can see these information

{C5742E99-9C23-4366-8AC7-EE8726F9EFFD}    2015-07-16 09:10:51:155+0700    1    147 [AGENT_DETECTION_FINISHED]    101    {00000000-0000-0000-0000-000000000000}    0    0    UpdateOrchestrator    Success    Software Synchronization    Windows Update Client successfully detected 3 updates.
{99634678-6C92-407C-B620-24901B2E3131}    2015-07-16 09:10:54:923+0700    1    147 [AGENT_DETECTION_FINISHED]    101    {00000000-0000-0000-0000-000000000000}    0    0    UpdateOrchestrator    Success    Software Synchronization    Windows Update Client successfully detected 1 updates.
{303601FF-C0A1-42D0-B9B7-4F9C3EA7A84D}    2015-07-16 09:10:55:073+0700    1    167 [AGENT_DOWNLOAD_STARTED]    101    {3F7B23C1-9339-44D4-81F9-8D7E82391671}    1    0    UpdateOrchestrator    Success    Content Download    Download started.
{A0DFCC85-4059-403F-AD77-35C44357ED68}    2015-07-16 09:11:03:287+0700    1    162 [AGENT_DOWNLOAD_SUCCEEDED]    101    {3F7B23C1-9339-44D4-81F9-8D7E82391671}    1    0    UpdateOrchestrator    Success    Content Download    Download succeeded.
{1AC03206-2DE8-46C7-A6BF-94A6528D1F50}    2015-07-16 09:11:03:304+0700    1    181 [AGENT_INSTALLING_STARTED]    101    {3F7B23C1-9339-44D4-81F9-8D7E82391671}    1    0    UpdateOrchestrator    Success    Content Install    Installation Started: Windows has started installing the following update: TH1 Professional 10240
{F756F686-9212-4C4C-B4F7-94FF722B6B7D}    2015-07-16 09:11:14:068+0700    1    182 [AGENT_INSTALLING_FAILED]    101    {3F7B23C1-9339-44D4-81F9-8D7E82391671}    1    8024200d    UpdateOrchestrator    Failure    Content Install    Installation Failure: Windows failed to install the following update with error 0x8024200d: TH1 Professional 10240.
{4533A9CF-1B34-40FC-BEB3-0854BB2402D8}    2015-07-16 09:11:14:474+0700    1    167 [AGENT_DOWNLOAD_STARTED]    101    {3F7B23C1-9339-44D4-81F9-8D7E82391671}    1    0    UpdateOrchestrator    Success    Content Download    Download started.
{E4F24968-BA89-444B-8535-8A50A3E176C9}    2015-07-16 09:11:16:626+0700    1    161 [AGENT_DOWNLOAD_FAILED]    101    {3F7B23C1-9339-44D4-81F9-8D7E82391671}    1    80200013    UpdateOrchestrator    Failure    Content Download    Error: Download failed.

If you even have tried these steps and it seems to be not working, we have another solution for you

  • Wired and wireless connections
  • Temporarily disabling anti-virus
  • Unchecking all proxy settings in Internet Explorer
  • Stopping Windows Update, removing the SoftwareDistribution folder, then rebooting
Here are the solution from our Microsoft Windows 10 expert:

Niranjan Manjhi, Microsoft  Support Engineer recommend you to follow these steps to make sure the “TH1 Professional 10240 – Error 0x80200013” will no longer happen.

1. I would suggest you to remove any external devices connected to the computer during installation process and check if that installs.
2. Run the DISM tool and check
Press Windows key + X and click Command Prompt (Admin) and type the following commands. Press the Enter key after each command:
 
DISM.exe /Online /Cleanup-image /Scanhealth 
DISM.exe /Online /Cleanup-image /Restorehealth

Another solution from Uttam Yadav, Microsoft  Forum Moderator:
Step 1. Perform Hardware clean boot.
You need to perform HARDWARE clean boot by removing external hardware’s like external Hard drive, printers, etc.
Follow this link which will help you to place the system in clean boot:

Note: Once you put the system in clean boot, please follow the step from that link to put the system in normal mode.
Step 2: Update the driver for Chipset, BIOS and Graphic card.
I suggest you to log in to manufacturer’s website and try updating the drivers to the latest drivers.
Step 3: Disable the security software completely.
If you are running any security software, i suggest you to disable it and check if it helps.
Important Note: Security software can help protect your computer against viruses and other security threats. In most cases, you shouldn’t disable your security software. If you have to temporarily disable it to install other software, you should re-enable it as soon as you’re done. If you’re connected to the Internet or a network while your antivirus software is disabled, your computer is vulnerable to attacks
If the issue persist then we need the CBS log files so that we can further investigate on this issue.
Please follow these steps to gather CBS log files:
a. From Start screen, type Command prompt.
b. Right-click on Command Prompt and choose Run as Adminstrator.
c. Now in the command prompt window,  type sfc /scannow (note, there is a space between sfc and /scannow)

If those steps does not help, please let us know via comment section below or download our useful tool to fix almost errors in Windows 10.