Section problem_knows_not_refl.
Variable town: t.
Variable n:ℕ.
Variable cardinality: | town | = 2×n+1.
Variable knows: elt ⇒ elt ⇒ Prop.
Infix "ℛ" := knows (at level 70, no associativity).
Variable knows_sym: ∀m n, m ℛ n ⇒ n ℛ m.
Variable knows_extensional:∀m n p, n≡p ⇒ m ℛ n⇒ m ℛ p.
Variable property: ∀B, B ⊆ town ⇒ |B| = n ⇒
∃d, d∈(town\B) /\ (∀b, b∈B ⇒ d ℛ b).

Lemma extendible_to_n:∀ B', B'⊆town ⇒ |B'| ≤ n ⇒ ∃B, |B| = n ∧  B'⊆B ∧ B⊆town.