The genitive construction τῆς Ἀντιμάχου is elliptical, relying on the implied noun (likely ὁμολογία, 'agreement' or 'pact', or perhaps a reference to a specific text or decree) rather than the explicit noun 'Antimachus'. This usage reflects the colloquial, compressed style of Aristophanic dialogue where the specific legal or contractual term is omitted because the context of the preceding argument (likely involving a dispute or a reference to a specific legal document) makes the meaning clear to the audience. The reference is to Antimachus of Colophon, a figure whose name might evoke a specific legal precedent or a well-known literary work in the Athenian cultural memory, though the exact nature of the 'agreement' or 'document' is left for the reader to infer from the immediate dramatic context.