HP has changed the 'helper' applications, Advisor and Assistant, packaged on their machines a few times over the past few years.
These links are a good reference to what those helper applications do, but the actual tool on your machine might be a different version. One of the pages I looked recommends updating to the newest version of the HP Software Assistant (advisor?) HPSA - I only did a quick scan to make sure the page applied to Win7 and notebooks.
Anyway, due diligence on your part is a good idea. Once you read more bit about the flashing BIOS process, you should feel more comfortable doing it. Read first, ask questions, and I'll try to answer them as best I can. I don't keep those helper apps on my machine, so I'll have to refer to and rely the HP documentation.
I'll look at the screen shot you posted earlier - but it would be good to know what version of the HP assistant is on your machine (it might be visible on the image you posted) ... it will definitely be in the help -> about section of the app.
Here are a few HP docs on the assistant, with references to flashing BIOS
If you launch HP Assistant, does it tell you anywhere when the BIOS was updaed?
Not the date of the BIOS itself, but when it was put on your machine.
Please understand that I'm not making any assertions, other than an incorrect BIOS version, or accusations below:
If the tech guys did it, you might take it back to them and say "Fix this, it is wrong!"
You can point to the HP support page if they argue it's a later version.
It's not, it's a different version that is not correct for your machine. In fact, it's a year older than the F.0A BIOS
The date it was put on your machine will tell you if they did flash it. If you can't determine the date the BIOS was flashed, then I don't think you'll have much success determining (proving) they updated BIOS.
You could always just ask them, then pursue their responsibility.
I don't know the shop, so I don't know if they're good guys - most shops are honest enough to admit when they made a mistake. Again, I don't know when the BIOS was flashed or who did it. The only relevant thing, from my perspective, is that is wrong and needs to be corrected in order to effectively troubleshoot any issue on the machine.
There's only one other thing that I have in my bag of tricks (I think - that's why I post things and say these are for later reference ... let me go over my own notes

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Bill
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