have heq : N0 = 0 := eq_bot_iff.mpr hlte
case intro.zero.inr
P : ℕ → Prop
h : ∀ (N : ℕ), P N → ∃ M, N < M ∧ P M
N0 : ℕ
hn0 : P N0
hlte : N0 ≤ 0
⊢ ∃ M, 0 < M ∧ P M
case intro.zero.inr
P : ℕ → Prop
h : ∀ (N : ℕ), P N → ∃ M, N < M ∧ P M
N0 : ℕ
hn0 : P N0
hlte : N0 ≤ 0
heq : N0 = 0
⊢ ∃ M, 0 < M ∧ P M