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  "assumptions": "Real normed vector spaces; f is an isometry equivalence between open balls with radii r > 0 and R > 0.",
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      "label": "MathlibAnnex.IsometryEquiv.existsUnique_affineExtension",
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      "text": "The proof or construction of Affine extension from open balls uses the project declaration “Mankiewicz extension on open connected domains” at the indicated step."
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  "conclusion": "f extends uniquely to an ambient real affine isometry equivalence. The two positive radii need not be assumed equal.",
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  "display_title": "Affine extension from open balls",
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  "name": "MathlibAnnex.IsometryEquiv.existsUnique_affineExtension_ball",
  "natural_language_statement": "Let f be a surjective isometry from the open ball ball(c,r) onto the open ball ball(d,R) in real normed spaces, where r>0 and R>0; the two radii need not be equal. Then there is a unique ambient real affine isometry equivalence A such that A(x)=f(x) for every x in the source ball.",
  "one_sentence_role": "Derives the open-ball form of the Mankiewicz extension theorem from the open-connected-domain theorem.",
  "proof_steps": [
    {
      "step_id": "step-1",
      "text": "Use positivity of R to record connectedness of the target ball; this fact is available but is not an additional hypothesis of the invoked extension theorem."
    },
    {
      "step_id": "step-2",
      "text": "Apply the open-connected extension theorem with the openness of both balls and the connectedness of the source ball."
    },
    {
      "step_id": "step-3",
      "text": "The existence, pointwise agreement, and uniqueness clauses are exactly those returned by that theorem."
    }
  ],
  "proof_summary": "Both balls are open, and the source ball is connected because its radius is positive. These facts put the isometry directly within the scope of the open-connected extension theorem.",
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    "text": "theorem existsUnique_affineExtension_ball\n    {c : E} {d : F} {r R : ℝ} (f : ball c r ≃ᵢ ball d R)\n    (hr : 0 < r) (hR : 0 < R) :\n    ∃! A : E ≃ᵃⁱ[ℝ] F,\n      ∀ x : ball c r, A (x : E) = ((f x : ball d R) : F)"
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