That should be sufficient. Since Office 2003 was installed before Office 2010, there should be no need for an Office 2010 reinstall on that account. Do note, however, that Microsoft says that Office 2003 (which is no longer supported) is not compatible with Windows 10 - notwithstanding that you may have been able to install it on that platform.
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Cheers,
Paul Edstein
[Fmr MS MVP - Word]
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