In mathematics one is often presented with a theorem, followed by a proof. At the end of the proof, one encounters commonly two phrases:
"We conclude the proof of the theorem..."
"We thus conclude the theorem".
Does this second sentence use the word 'conclude' properly? I believe it does, though it is obviously in a slightly different sense than the first. The first sentence says "We end the proof", while the second says "We judge the theorem to be true".
I would appreciate a clarification of this.