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Yep you should be able to use that coaxial outlet and run it to another room. Of course you'll have to tune the TV to the outputted frequency from the STB. You will only be able to watch one STB transmission at a time i.e. same thing downstairs as upstairs.
If you are using the coaxial, then all you need is a means of transmitting the remote signal when you are upstairs to the STB downstairs. A video sender would duplicate or replace the coaxial lead.
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