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  "assumptions": "Real normed vector spaces; f is an isometry equivalence s to t; radius-r symmetric lenses about x,y and f(x),f(y) lie in s and t; half the distance from x to y is at most r.",
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      "text": "The proof or construction of Midpoint preservation under lens containment uses the project declaration “Midpoint preservation on a symmetric lens” at the indicated step."
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      "text": "The proof or construction of Midpoint preservation under lens containment uses the project declaration “Restriction of a set isometry to corresponding symmetric lenses” at the indicated step."
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      "text": "The proof or construction of Midpoint preservation under lens containment uses the project declaration “The midpoint belongs to a sufficiently large symmetric lens” at the indicated step."
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  "conclusion": "f preserves the midpoint of x and y. The target half-distance bound follows from the isometry.",
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  "display_title": "Midpoint preservation under lens containment",
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  "name": "MathlibAnnex.IsometryEquiv.map_midpoint_of_symmetricLens_subset",
  "natural_language_statement": "Let f:s≃ᵢt and x,y∈s. If the symmetric lens of radius r around x,y lies in s, the corresponding lens around f(x),f(y) lies in t, and half the distance between x and y is at most r, then f sends midpoint(x,y) to midpoint(f(x),f(y)).",
  "one_sentence_role": "Applies the symmetric-lens midpoint theorem inside arbitrary source and target subsets.",
  "proof_steps": [
    {
      "step_id": "step-1",
      "text": "Construct the induced isometry equivalence between the two symmetric lenses."
    },
    {
      "step_id": "step-2",
      "text": "Use f.dist_eq to identify the distance between the target foci with the distance between the source foci."
    },
    {
      "step_id": "step-3",
      "text": "Transport the half-distance hypothesis to the target lens."
    },
    {
      "step_id": "step-4",
      "text": "Apply midpoint preservation for an isometry between the two lenses."
    }
  ],
  "proof_summary": "Restrict f and its inverse to the two contained lenses. Distance preservation transports the half-distance bound to the target foci. The lens-level midpoint theorem applied to this restricted isometry gives the result.",
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    "text": "theorem map_midpoint_of_symmetricLens_subset\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t) (x y : s) (r : ℝ)\n    (hhalf : 2⁻¹ * dist (x : E) (y : E) ≤ r)\n    (hs : symmetricLens (x : E) (y : E) r ⊆ s)\n    (ht : symmetricLens ((f x : t) : F) ((f y : t) : F) r ⊆ t) :\n    ((f ⟨midpoint ℝ (x : E) (y : E),\n          hs (midpoint_mem_symmetricLens hhalf)⟩ : t) : F) =\n      midpoint ℝ ((f x : t) : F) ((f y : t) : F)"
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