Dangerous Liaisons

In 1781 Paris, the Marquise de Merteuil plots revenge against her ex-lover, the Comte de Bastide. To soothe her wounded pride, she seeks to arrange the seduction and disgrace of his virgin fiancÃ©e CÃ©cile de Volanges, who has recently been presented to society after spending her formative years in the shelter of a convent.
Merteuil calls on her sometime partner, the Vicomte de Valmont, and asks him to seduce CÃ©cile. Valmont declines as he is currently trying to seduce Madame de Tourvel, the wife of a presiding justice who is staying with his wealthy aunt Madame de Rosemonde while her husband is away on a court case.
After some persuasion, Valmont reluctantly agrees to ruin CÃ©cile and destroy Bastide's reputation on the condition that Merteuil modify the proposal: should he successfully seduce Tourvel, Merteuil must share her bed with him. Merteuil accepts, but asks that Valmont furnish written proof of the liaison.
While visiting Rosemonde's country estate, Valmont attempts to attract Tourvel's attention only to have his advances rebuffed. Valmont then instructs his page to seduce Tourvel's maid and gain access to her private correspondence.
Upon learning that CÃ©cile's mother, Madame de Volanges, has warned Tourvel of his untrustworthy nature, Valmont resolves to seduce CÃ©cile as revenge for her mother's denunciation.
Meanwhile, CÃ©cile develops a relationship with her music teacher Raphael Danceny, unaware that Merteuil introduced him to CÃ©cile knowing that Danceny came from a poor background and therefore does not qualify in the eyes of her mother as a potential suitor.
Valmont's seduction of CÃ©cile is rapid and unsubtle: After gaining access to her bedchamber on a false pretense, he forces himself upon her as she pleads with him to leave. On the pretext of illness CÃ©cile remains locked in her chambers, refusing all visitors. A concerned Madame de Volanges calls upon Merteuil to speak to her. CÃ©cile, naively assuming that Merteuil has her best interests at heart, confides in her. Merteuil advises CÃ©cile to welcome Valmont's advances; young women should take advantage of all the lovers they can acquire, she says, in a society so repressive and contemptuous of women. The result is a perverse "student-teacher" relationship between CÃ©cile and Valmont; by day she is courted by Danceny, and each night she receives a sexual "lesson" from Valmont. In the meantime, Merteuil begins an affair with Danceny.
Meanwhile, during Valmontâs time as CÃ©cile's "teacher", he somehow manages to win Tourvel's heart â but at a cost: Valmont, the lifelong bachelor playboy, falls in love. In a fit of jealousy, Merteuil mocks Valmont and threatens to trash his reputation as a carefree gigolo. She also refuses to honor her end of their agreement, since Valmont has no written proof that the relationship has been consummated. Valmont abruptly dismisses Tourvel with a terse excuse: "It is beyond my control". CÃ©cile, meanwhile, after a particularly rough night in Valmont's bed, miscarries his child.
Overwhelmed with grief and shame, Tourvel retires to a convent where her health deteriorates rapidly while a guilt-ridden CÃ©cile returns to the convent from whence she came. When Valmont warns Danceny of Merteuil's ulterior motives, Merteuil retaliates by informing Danceny that Valmont has been sleeping with CÃ©cile. Danceny challenges Valmont to a duel and mortally wounds him.
With his dying breath, Valmont asks Danceny to visit Tourvel and tell her their love was real before giving him the letters proving Merteuil's involvement. Danceny does so and publishes the letters, revealing Merteuil's schemes to everyone in Paris. After being booed by her former friends and sycophants at the opera, Merteuil flees the city in disgrace.

Question: What does Valmont have to furnish proof of?
Answer: The liason