have r3 := y_mod_three (unit ^ (3 * b) : ℤθ).h 0 t3
case left.succ.intro.intro.intro.intro
a b✝ c : ℤθ
b : ℕ
h1 : (↑(unit ^ (3 * ↑b))).f % 3 = 1
h2 : (↑(unit ^ (3 * ↑b))).g % 3 = 0
h3 : (↑(unit ^ (3 * ↑b))).h % 3 = 0
p : Nat.succ b = b + 1
w : ↑(unit ^ (3 * (b + 1))) = ↑(unit ^ (3 * b)) * ↑unit ^ 3
t1 : (↑(unit ^ (3 * b))).f % 3 = 1
t2 : (↑(unit ^ (3 * b))).g % 3 = 0
t3 : (↑(unit ^ (3 * b))).h % 3 = 0
c1 : ℤ
hc1 : (↑(unit ^ (3 * b))).f = 3 * c1 + 1
c2 : ℤ
hc2 : (↑(unit ^ (3 * b))).g = 3 * c2
⊢ (↑(unit ^ (3 * (b + 1)))).f % 3 = 1 ∧ (↑(unit ^ (3 * (b + 1)))).g % 3 = 0 ∧ (↑(unit ^ (3 * (b + 1)))).h % 3 = 0
case left.succ.intro.intro.intro.intro
a b✝ c : ℤθ
b : ℕ
h1 : (↑(unit ^ (3 * ↑b))).f % 3 = 1
h2 : (↑(unit ^ (3 * ↑b))).g % 3 = 0
h3 : (↑(unit ^ (3 * ↑b))).h % 3 = 0
p : Nat.succ b = b + 1
w : ↑(unit ^ (3 * (b + 1))) = ↑(unit ^ (3 * b)) * ↑unit ^ 3
t1 : (↑(unit ^ (3 * b))).f % 3 = 1
t2 : (↑(unit ^ (3 * b))).g % 3 = 0
t3 : (↑(unit ^ (3 * b))).h % 3 = 0
c1 : ℤ
hc1 : (↑(unit ^ (3 * b))).f = 3 * c1 + 1
c2 : ℤ
hc2 : (↑(unit ^ (3 * b))).g = 3 * c2
r3 : ∃ k, (↑(unit ^ (3 * b))).h = 3 * k + 0
⊢ (↑(unit ^ (3 * (b + 1)))).f % 3 = 1 ∧ (↑(unit ^ (3 * (b + 1)))).g % 3 = 0 ∧ (↑(unit ^ (3 * (b + 1)))).h % 3 = 0