English
If certain Galois connections hold, then the two-variable image infimum equals a supremum-infimum mixture of the bounds.
Русский
При выполнении некоторых Гало-соединений образ2 равен смеси верхних и нижних пределов.
LaTeX
$$$(h_1,h_2): s.Nonempty \Rightarrow BddBelow s \Rightarrow t.Nonempty \Rightarrow BddAbove t \Rightarrow sInf (image2 l s t) = l (sInf s) (sSup t)$$$
Lean4
theorem csInf_image2_eq_csInf_csSup (h₁ : ∀ b, GaloisConnection (l₁ b) (swap u b))
(h₂ : ∀ a, GaloisConnection (toDual ∘ l₂ a) (u a ∘ ofDual)) :
s.Nonempty → BddBelow s → t.Nonempty → BddAbove t → sInf (image2 u s t) = u (sInf s) (sSup t) :=
csInf_image2_eq_csInf_csInf (β := βᵒᵈ) h₁ h₂