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Hello, With every TV that I know of, there is no way to program the TV's original remote to control any sort of cable box/mini-box, SAT box, DTV tuner box, etc. The only way to typically do this is to use the cable box included remote and gain "basic" functionality of the TV. However there are universal remotes that can be used. However I don't have much information to go off of here.
Can you postback with:
1) The model number off of the back of the TV?
2) The brand and model off of the front of that "Mini-Box"?
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Do you need further assistance with this?