English
Let s ⊆ α and t ⊆ β be sets and f: α × β → γ satisfy that f is antitone in the first argument and antitone in the swapped second argument. Then for any a ∈ lowerBounds s and b ∈ lowerBounds t, the value f(a,b) is an upper bound of all values f(x,y) with x ∈ s and y ∈ t. Equivalently, image2 f (lowerBounds s) (lowerBounds t) ⊆ upperBounds (image2 f s t).
Русский
Пусть s ⊆ α и t ⊆ β. Пусть f: α × β → γ удовлетворяет условиям антимонотонности по каждому аргументу. Тогда для любых a ∈ нижних границ s и b ∈ нижних границ t значение f(a,b) является верхней границей множества { f(x,y) : x ∈ s, y ∈ t }. То есть image2 f (lowerBounds s) (lowerBounds t) ⊆ upperBounds (image2 f s t).
LaTeX
$$$image2 f (\\lowerBounds s) (\\lowerBounds t) \\subseteq \\ upperBounds (\\image2 f s t)$$$
Lean4
theorem image2_upperBounds_upperBounds_subset_upperBounds_image2 :
image2 f (lowerBounds s) (lowerBounds t) ⊆ upperBounds (image2 f s t) :=
image2_subset_iff.2 fun _ ha _ hb ↦ mem_upperBounds_image2_of_mem_lowerBounds h₀ h₁ ha hb
