occupied / topped / proven
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occupied
Occupied or reserved, so that something is not available. This can refer to a seat, a room, a course, or a phone line.
Leider sind schon alle Plätze im Deutschkurs belegt.
Unfortunately, all the places in the German course are already occupied.
'Belegt' (occupied / topped / proven) is the past participle of the verb 'belegen' (to occupy / to top / to prove) and is used here as an adjective. The different meanings of 'belegt' (a seat is occupied, a piece of bread is topped) come from the different meanings of the verb 'belegen'.
Der Platz neben mir ist leider schon belegt.
The seat next to me is unfortunately already occupied.
You can recognize the meaning of 'belegt' (occupied / topped / proven) from the noun it describes. A 'belegte Stimme' (hoarse voice) is hoarse, a 'belegter Platz' (occupied seat) is taken, and a 'belegtes Brot' (topped/open sandwich) is covered with something.
Nach der langen Rede war seine Stimme ganz belegt.
After the long speech, his voice was completely hoarse.
When 'belegt' (topped) is used for food, it usually describes bread or rolls. With the preposition 'mit' (with), you can specify what is on top.
Für die Reise haben wir uns Brötchen mit Wurst und Käse belegt.
For the trip, we made ourselves rolls topped with sausage and cheese.
In the meaning 'nachgewiesen' (proven), 'belegt' (proven) is often used with the preposition 'durch' (by/through) to name the source of the evidence. This use is typical of formal or scientific texts.
Seine Behauptung ist durch keine Fakten belegt.
His claim is not proven by any facts.
The combination 'eine belegte Stimme haben' (to have a hoarse voice) is a fixed expression and means that you are hoarse.
Am Morgen nach der Party hatte ich eine total belegte Stimme.
The morning after the party, I had a really hoarse voice.
free
'frei' and 'belegt' are opposites when talking about seats, rooms, courses, or lines. 'frei' means that something is available. 'belegt' means that it is already being used, booked, or assigned. Learners may confuse 'belegt' with other meanings, for example in 'ein belegtes Brot' (an open-faced sandwich). But this relation is only about availability. So the question is: is there still a place 'frei', or is everything 'belegt'?
Der Deutschkurs ist leider schon belegt.
Unfortunately, the German course is already full.
Im Bus ist vorne noch ein Sitz frei.
There is still a seat available at the front of the bus.