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Starting from Wightman axioms, we can define the Schwinger functions as the Wick-rotated Wightman functions (as for instance is explained in the book by R. Haag, Local Quantum Physics).

The Schwinger functions have a set of properties, which essentially come from the axioms of the original, lorentzian theory. In particular, Schwinger functions are analytic away from coincident points, as claimed in the aforementioned book.

How can we prove this claim? The book by Haag doesn't seem to explain this.

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If your starting point is a Wightman theory then you can find a proof in Section II.3 of the book "The $P(\Phi)_2$ Euclidean (Quantum) Field Theory" by Simon. If your starting point is a given set of Schwinger functions satisfying the Osterwalder-Schrader axioms then an alternative derivation is in the book "Quantum Physics: A Functional Integral Point of View" by Glimm and Jaffe, more precisely Corollary 19.5.6 (in the Second Edition).

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