specialize this a h'.1
case a.mpr
t : ℕ
P : ℕ → ℕ
h : Balanced t P
this : ∀ (i : ℕ), i ∈ range (t + 1) → P i = minP t P ∨ P i = minP t P + 1
a : ℕ
h' : a ∈ range (t + 1) ∧ ¬P a = minP t P
⊢ P a = minP t P + 1
case a.mpr
t : ℕ
P : ℕ → ℕ
h : Balanced t P
a : ℕ
h' : a ∈ range (t + 1) ∧ ¬P a = minP t P
this : P a = minP t P ∨ P a = minP t P + 1
⊢ P a = minP t P + 1