obtain ⟨⟨a, ha, ha₂⟩, b, hb, hb₂⟩ := h (Submonoid.mem_prod.2 ⟨hxy, t.one_mem⟩)
M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : Monoid N
s : Submonoid M
t : Submonoid N
x y : M
hxy : x * y ∈ s.toSubsemigroup
h :
  (x * y, 1) ∈ (prod s t).toSubsemigroup →
    (∃ z ∈ (prod s t).toSubsemigroup, Associated (x, 1) z) ∧ ∃ z ∈ (prod s t).toSubsemigroup, Associated (y, 1) z
⊢ (∃ z ∈ s.toSubsemigroup, Associated x z) ∧ ∃ z ∈ s.toSubsemigroup, Associated y z
case intro.intro.intro.intro.intro
M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : Monoid N
s : Submonoid M
t : Submonoid N
x y : M
hxy : x * y ∈ s.toSubsemigroup
h :
  (x * y, 1) ∈ (prod s t).toSubsemigroup →
    (∃ z ∈ (prod s t).toSubsemigroup, Associated (x, 1) z) ∧ ∃ z ∈ (prod s t).toSubsemigroup, Associated (y, 1) z
a : M × N
ha : a ∈ (prod s t).toSubsemigroup
ha₂ : Associated (x, 1) a
b : M × N
hb : b ∈ (prod s t).toSubsemigroup
hb₂ : Associated (y, 1) b
⊢ (∃ z ∈ s.toSubsemigroup, Associated x z) ∧ ∃ z ∈ s.toSubsemigroup, Associated y z