Blanche and Stanley have their first big confrontation in Scene 2. Who wins this round? What does Blanche do to disarm Stanley?

Blanche and Stanley have their first argument about Blanche's claim that Belle Reve has been lost. Stanley seems to be winning the argument when Blanche changes tactics and seems to tell him the truth, rather than flirting and telling lies. Blanche disarms Stanley when she produces lawyer's papers, after which Stanley calms down. Note, Stanley does say he will be showing the papers to a lawyer in the future.