Step 1: Brown-Sequard syndrome results from hemisection (damage to one half) of the spinal cord, producing a characteristic dissociated sensory loss.
Step 2: The dorsal column-medial lemniscus pathway carries proprioception, fine touch and vibration. These fibres ascend on the same side of the cord and only cross in the medulla. So below the lesion, proprioception is lost on the same (ipsilateral) side as the injury, along with ipsilateral motor paralysis.
Step 3: The spinothalamic tract carries pain and temperature. These fibres cross within one to two segments of entering the cord, so pain and temperature are lost on the opposite (contralateral) side. This is why Pain (a) and Temperature (d) are wrong for an ipsilateral loss.
Step 4: Hence the sensation lost on the same side as the lesion is proprioception, making option c correct.