Can the generalized Haag theorem be further generalized

$L^{\uparrow }$-invariant n-point functions of scalar field theories satisfying the Wightman axioms are considered in the framework of the recently proposed inhomogeneous U(3,1)-invariant extension which is Weierstrass analytic in both the real and imaginary parts of complex four-vectors. The algebraic variety over which the extension is analytic is investigated, and it is shown that there is a shift in the appearance of singular points from n≥6 as for the customary complex analytic extension, to n≥10. The extended analyticity domain is investigated too, and it is proved that it contains all the spacelike points of the analyticity domain of the physical n-point function. A procedure to reach physical timelike separation, as well as any separation, is introduced, and it is shown that the above type of Weierstrass analyticity is sufficient to determine the physical n-point function at any separation from its value at spacelike separation. The above results are applied to the generalized Haag theorem in order to see whether its validity can be extended to more than the first four vacuum expectation values for the considered type of field theories.