If b < c If b = 0 c > 1 ∧ c | 0 ∧ c | c [c|o lemma] c > 1 ∧ c | b ∧ c | c ∃d∈ℕ ( d > 1 ∧ d | b ∧ d | c ) ¬∃d∈ℕ ( d > 1 ∧ d | b ∧ d | c ) ⊥ b > 0 b ≥ 1 b.c ≥ c ∃x ∈ ℕ (b.x ≥ c) ∃m ∈ ℕ (b.m ≥ c ∧ ∀k∈ℕ ( b.k ≥ c ⇒ k≥m ) ). Let u ∈ ℕ such that b.u ≥ c b.u ≥ c Let p ∈ ℕ such that b.u = c + p If u = 0 b.u ≥ c 0 ≥ c 0 ≥ c > 1 0 > 1 ⊥ u > 0 u ≥ 1