Flash Media Live Encoder(FMLE) does not have the capability to broadcast pre-recorded video. The feed comes straight from your webcam, so whatever is streaming from the FMLE, is coming from your webcam.
However, you may want to look into Wirecast, which is similar to FMLE. With Wirecast, you can stream pre-recorded video. While Wirecast is a paid service, it does offer a free service, but you do not have access to all the features, but should be fine if you plan to stream only one video at a time.
There are other things you can do to 'mask' your stream to make it seem like it is Live and not a pre-recorded broadcast. In your broadcast player, you can remove some of the features that make is seem pre-recorded instead of live, like the pause button and the scroll bar.
I hope this helps, and if you have any further questions, please let me know!