So, Fe covers all of Fi? Not exactly. On the outside, yes. But from the observer's perspective? No. The missing element? Direct experience of the emotion by the observer. It's felt by the others, and can be internally confirmed by the observer via their reflections and ideas, even written, but never seen directly on the internal level. We can easily know what we are feeling, and are driven to know. But we can only know, not feel. The advantage? Awareness of what others are feeling. Does Fi exist in Fe's? Technically yes. But we aren't directly aware of it. The Intuitive is luckier, though. Intuition acts as the best mirror.
How do we know that waves exist outside the visible spectrum if we can't see them? By the effect they have on objects attuned to their presence.
How did Newton discover gravity? Did he see it with his eyes? No. He saw the apple fall, and inferred gravity from the impact it had upon the apple.
The reason we constantly output emotion is in hopes that another will reflect it back. Holding a forward-directed flashlight, desperate for mirrors, but all the mirrors are slightly flawed, curved, or dim. So we need several to get a glimpse of ourselves.
So, Fe is feeling projected onto the object, reflected back. What you believed covered Fi was only the reflection of it.