someone can help with the passage marked in red. I've tried in many ways using the definition, but to no avail. This theorem is found in Bartle's book The Elements of Integration and Lebesgue measure on page 142. Definition 13.3 that is needed at this beginning of the demonstration and the condition of Carathéodory $m^*(A)= m^*(A \cap E)+m^*(A \cap E^c)$.
Just use the De Morgan's Laws.