A cut-free calculus for second-order Gödel logic

We prove that the extension of the known hypersequent calculus for standard first-order Gödel logic with usual rules for second-order quantifiers is sound and (cut-free) complete for Henkin-style semantics for second-order Gödel logic. The proof is semantic, and it is similar in nature to Schütte and Tait's proof of Takeuti's conjecture.