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  "assumptions": "Real normed vector spaces; the source s is convex with nonempty interior; f is an isometry equivalence s to t; x is interior to s exactly when f(x) is interior to t.",
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      "label": "MathlibAnnex.IsometryEquiv.existsUnique_affineExtension",
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      "text": "The proof or construction of Convex-set extension via ambient interiors uses the project declaration “Mankiewicz extension on open connected domains” at the indicated step."
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      "text": "The proof or construction of Convex-set extension via ambient interiors uses the project declaration “Restriction of an isometry to ambient interiors” at the indicated step."
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  "conclusion": "f extends uniquely to an ambient real affine isometry equivalence. No additional convexity assumption on t is stated.",
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  "name": "MathlibAnnex.IsometryEquiv.existsUnique_affineExtension_of_convex",
  "natural_language_statement": "Let f be a surjective isometry from a convex subset s of a real normed space onto a subset t. Assume the ambient interior of s is nonempty and, for every x in s, x lies in interior(s) if and only if f(x) lies in interior(t). Then f extends uniquely to an ambient real affine isometry equivalence. No separate convexity assumption on t is required by this statement.",
  "one_sentence_role": "Transfers the open-connected extension theorem to convex sets whose ambient interiors are matched exactly by the isometry.",
  "proof_steps": [
    {
      "step_id": "step-1",
      "text": "Construct an isometry equivalence between interior(s) and interior(t) from the bidirectional interior-membership hypothesis."
    },
    {
      "step_id": "step-2",
      "text": "Apply the open-connected extension theorem to the two open interiors, using connectedness of the source interior obtained from convexity of s."
    },
    {
      "step_id": "step-3",
      "text": "Identify the subtype of points of s lying in interior(s) and prove it is dense in s via the convex closure-of-interior theorem."
    },
    {
      "step_id": "step-4",
      "text": "Use continuity of the affine extension and of f to extend their equality from the dense interior subtype to every point of s."
    },
    {
      "step_id": "step-5",
      "text": "For uniqueness, restrict any competing ambient affine isometry to the source interior and invoke the uniqueness clause from the open-connected theorem."
    }
  ],
  "proof_summary": "Restrict f to an isometry between the two ambient interiors. Convexity and nonempty interior make the source interior connected, so the open-connected theorem yields an affine extension on the interiors. The interior of a convex set with nonempty interior is dense in the set; continuity therefore extends agreement to all of s. Uniqueness is inherited by restriction to the interior.",
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    "text": "theorem existsUnique_affineExtension_of_convex\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t)\n    (hs : Convex ℝ s) (hsint : (interior s).Nonempty)\n    (hfi : ∀ x : s,\n      (x : E) ∈ interior s ↔ ((f x : t) : F) ∈ interior t) :\n    ∃! A : E ≃ᵃⁱ[ℝ] F,\n      ∀ x : s, A (x : E) = ((f x : t) : F)"
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