(1) a. They proved there to be an error in the calculation.
b. There proved to be an error in the calculation.
B. Accusative/objective case is checked on there in (1a) and nominative case in (1b) because there is the subject of the infinitival complement to an ECM verb prove in (1a) and the subject of a finite clause in (1b).
C. In (1a) case on there is checked in the head-spec configuration. In (1b) it is checked in the spec-head configuration.
D. In (1a) the agent of prove is present in subject position because the sentence is in the active voice. By Burzio's generalization and the argument structure of the verb, the verb has an objective case to license. Because prove is an ECM verb this case licensing applies to the subject of the complement infinitive. In (1b) the passive form of the verb is used and by Burzio's generalization, the subject disappears along with the licensing of objective case. As a result, the subject position of the main clause is empty and the subject of the infinitive must move to this position to get its case licensed.