Is it possible to check a bag through to the destination of a different itinerary?

I need to check bags from a flight (intinerary ABC) going to Paris on Austrian Airlines purchased from one travel company (comp X) to my follow on flight (itinerary DEF) purchased from a different company (comp Y) on Air France in Paris. Is there any way that I can get the bags to follow me through the baggage system so that I do not have to go through customs in Paris to get my checked bags from Austrian only to go recheck them again with Air France?

Is it possible that when I check in for itinerary ABC with Austrian to have my luggage also tagged with my separate itinerary with Air France itinerary DEF so that I do not need to retrieve my bags in Paris and go back through security and the check in process again? I want to be able to board my flight in Cairo Egypt and have my bags on the Austrian flight follow me through my layovers in Vienna, Paris and then moved to Air France on a separate itinerary to Atlanta and to my home in San Antonio Texas. Is this possible or will I need to do the whole customs, immigration and flight check in non-sense again in Paris when I finish one itinerary and start the next. (These flights are all going to take place in the span of about 19 hours or so.)