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A supergravity extension of the (R+R2) gravity with the additional (Born-Infeld) structure of a massive vector multiplet gives rise to the specific F(R) gravity, whose structure is investigated in detail. The massive vector multiplet has an inflaton (scalaron), goldstino, and massive vector field as its field components. The model describes Starobinsky inflation and allows us to extrapolate the F(R) function beyond the inflationary scale (up to Planck scale). We observe some differences versus the (R+R2)gravity and several breaking patterns of the well-known correspondence between the F(R) gravity and the scalar-tensor gravity.