obtain ⟨⟨a, b, c⟩, (habc : SimpleTriangle a b c), hmin⟩ :=
  S.toFinite.exists_minimal_wrt (fun ⟨x, y, z⟩ => Delta x y z) S this
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, SimpleTriangle x y z
S : Set (V × V × V) := {(x, y, z) | SimpleTriangle x y z}
this : S.Nonempty
⊢ ∃ a b, a ≠ b ∧ Line a b = {a, b}
case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, SimpleTriangle x y z
S : Set (V × V × V) := {(x, y, z) | SimpleTriangle x y z}
this : S.Nonempty
a b c : V
habc : SimpleTriangle a b c
hmin :
  ∀ a' ∈ S,
    ((match a' with
        | (x, y, z) => Delta x y z) ≤
        match (a, b, c) with
        | (x, y, z) => Delta x y z) →
      (match (a, b, c) with
        | (x, y, z) => Delta x y z) =
        match a' with
        | (x, y, z) => Delta x y z
⊢ ∃ a b, a ≠ b ∧ Line a b = {a, b}