English
The extended chart on a composition of charts is the composition of the corresponding charts, i.e., extChartAt I x equals the transitive composition of charts.
Русский
Расширенная карта на композицию карт равна композиции соответствующих карт.
LaTeX
$$$extChartAt I x = (chartAt H' x) \toPartialEquiv \circ extChartAt I (chartAt H' x x)$$$
Lean4
theorem extChartAt_prod (x : M × M') : extChartAt (I.prod I') x = (extChartAt I x.1).prod (extChartAt I' x.2) :=
by
simp only [mfld_simps]
rw [PartialEquiv.prod_trans]