English
For index i in I, the gauge of the i-th coordinate set is bounded by the gauge of the product set.
Русский
Для индекса i в I гейдж i-й координатной множества ограничен гейджем произведения.
LaTeX
$$$\forall i \in I, \mathrm{egauge}_{\mathbb{K}}(s_i)(x_i) \le \mathrm{egauge}_{\mathbb{K}}(\Pi_I s)(x)$$$
Lean4
theorem egauge_union (s t : Set E) (x : E) : egauge 𝕜 (s ∪ t) x = egauge 𝕜 s x ⊓ egauge 𝕜 t x :=
by
unfold egauge
simp [smul_set_union, iInf_or, iInf_inf_eq]