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  "assumptions": "Real normed vector spaces; an isometry equivalence between symmetric lenses of the same radius r; both half-distances between the respective foci are at most r.",
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      "label": "_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.mapsTo_pointReflection_symmetricLens",
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      "text": "The proof or construction of Midpoint preservation on a symmetric lens uses the project declaration “A symmetric lens is invariant under reflection about its midpoint” at the indicated step."
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      "text": "The proof or construction of Midpoint preservation on a symmetric lens uses the project declaration “The midpoint belongs to a sufficiently large symmetric lens” at the indicated step."
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  "conclusion": "The isometry maps the midpoint of the source foci to the midpoint of the target foci.",
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  "display_title": "Midpoint preservation on a symmetric lens",
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  "name": "MathlibAnnex.IsometryEquiv.map_midpoint_of_symmetricLens",
  "natural_language_statement": "Assume the midpoint of x,y belongs to symmetricLens(x,y,r), and likewise the midpoint of x′,y′ belongs to symmetricLens(x′,y′,r), as expressed by the two half-distance inequalities. Any isometry equivalence between these lenses maps midpoint(x,y) to midpoint(x′,y′).",
  "one_sentence_role": "Turns bounded reflection symmetry of equal-radius lenses into midpoint preservation.",
  "proof_steps": [
    {
      "step_id": "step-1",
      "text": "Use the half-distance inequalities to place both midpoints in their respective lenses."
    },
    {
      "step_id": "step-2",
      "text": "Bound the source lens by one of its defining closed balls."
    },
    {
      "step_id": "step-3",
      "text": "Use reflection invariance of each lens about the corresponding midpoint."
    },
    {
      "step_id": "step-4",
      "text": "Apply center transport to the lens isometry and erase the subtype wrapper."
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  "proof_summary": "A symmetric lens is bounded and invariant under point reflection about the midpoint of its foci. The center-transport theorem therefore sends the source midpoint to the target midpoint.",
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    "text": "theorem map_midpoint_of_symmetricLens\n    {x y : E} {x' y' : F} {r : ℝ}\n    (hxy : 2⁻¹ * dist x y ≤ r) (hxy' : 2⁻¹ * dist x' y' ≤ r)\n    (f : symmetricLens x y r ≃ᵢ symmetricLens x' y' r) :\n    ((f ⟨midpoint ℝ x y, midpoint_mem_symmetricLens hxy⟩ :\n        symmetricLens x' y' r) : F) = midpoint ℝ x' y'"
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