English
The chart at x for the composed space equals the composition of charts: chartAt H x = chartAt H' x trans chartAt H (chartAt H' x x).
Русский
Карта chartAt для композиции равна композиции карт: chartAt H x = chartAt H' x ∘ chartAt H (chartAt H' x x).
LaTeX
$$chartAt H x = (chartAt H' x) .trans (chartAt H (chartAt H' x x))$$
Lean4
theorem chartAt_comp (H : Type*) [TopologicalSpace H] (H' : Type*) [TopologicalSpace H'] {M : Type*}
[TopologicalSpace M] [ChartedSpace H H'] [ChartedSpace H' M] (x : M) :
(letI := ChartedSpace.comp H H' M;
chartAt H x) =
chartAt H' x ≫ₕ chartAt H (chartAt H' x x) :=
rfl