What is the ruling when both parties possess proof corresponding to their respective claims?

Primary text

When each party has proof supporting their specific claim, the proofs contradict each other. The half claimed by the party claiming the entirety is secured for him. The other half is subject to the disagreement concerning which proof takes precedence. The apparent position of the Madhhab is to prioritize the proof of the claimant (of the entirety), resulting in the entire house being awarded to the claimant of the whole. This is the position of Abu Hanifa and his two companions.