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  "assumptions": "Real normed vector spaces; radius-R balls about c and f(c) lie in the source and target; R > 0; x,y lie in the radius-R/4 source ball.",
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      "label": "MathlibAnnex.IsometryEquiv.map_midpoint_of_symmetricLens_subset",
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      "text": "The proof or construction of Midpoint preservation on a quarter ball uses the project declaration “Midpoint preservation under lens containment” at the indicated step."
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      "text": "The proof or construction of Midpoint preservation on a quarter ball uses the project declaration “Symmetric lens” at the indicated step."
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  "conclusion": "f sends the midpoint of x and y to the midpoint of f(x) and f(y).",
  "declaration_kind": "theorem",
  "definition_explanation": null,
  "display_title": "Midpoint preservation on a quarter ball",
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  "name": "MathlibAnnex.IsometryEquiv.map_midpoint_of_mem_ball",
  "natural_language_statement": "Let f:s≃ᵢt and let c∈s. Assume R>0, ball(c,R)⊆s, and ball(f(c),R)⊆t. If x,y∈s both lie in ball(c,R/4), then f sends their midpoint to the midpoint of f(x) and f(y).",
  "one_sentence_role": "Obtains a uniform local midpoint law from symmetric-lens center rigidity.",
  "proof_steps": [
    {
      "step_id": "step-1",
      "text": "Use the triangle inequality through c to bound dist(x,y) by less than R/2."
    },
    {
      "step_id": "step-2",
      "text": "Deduce the half-distance condition needed for a symmetric lens of radius R/2."
    },
    {
      "step_id": "step-3",
      "text": "Show every point of the source lens lies in ball(c,R) by combining its distance to x with the R/4 bound on x."
    },
    {
      "step_id": "step-4",
      "text": "Transport the R/4 bound from x to f(x), then prove the analogous containment of the target lens in ball(f(c),R)."
    },
    {
      "step_id": "step-5",
      "text": "Apply midpoint preservation for a set isometry whose corresponding lenses remain inside the source and target sets."
    }
  ],
  "proof_summary": "The two endpoints are less than R/2 apart. Their symmetric lens of radius R/2 lies in the source R-ball, and the corresponding target lens lies in the target R-ball. The subset-local lens theorem therefore applies.",
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    "text": "theorem map_midpoint_of_mem_ball\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t) (c : s) {R : ℝ} (hR : 0 < R)\n    (hsball : ball (c : E) R ⊆ s)\n    (htball : ball ((f c : t) : F) R ⊆ t)\n    (x y : s) (hx : (x : E) ∈ ball (c : E) (R / 4))\n    (hy : (y : E) ∈ ball (c : E) (R / 4)) :\n    ((f ⟨midpoint ℝ (x : E) (y : E),\n          hsball (by\n            rw [mem_ball]\n            calc\n              dist (midpoint ℝ (x : E) (y : E)) (c : E) ≤\n                  (dist (x : E) (c : E) + dist (y : E) (c : E)) / 2 := by\n                    simpa [dist_comm] using\n                      dist_midpoint_midpoint_le (x : E) (y : E) (c : E) (c : E)\n              _ < R := by rw [mem_ball] at hx hy; linarith)⟩ : t) : F) =\n      midpoint ℝ ((f x : t) : F) ((f y : t) : F)"
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