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_by_one:∀ B', |B'| ≤ |town| -1 ⇒ ∃d, d∈town ∧ ¬(d∈B').