In the book differential geometry of complex vector bundles by Kobayashi, the book is free online, https://www.mathsoc.jp/assets/pdf/publications/pubmsj/Vol15.pdf. The screenshot of the page 5 is below.
In the proof of the equivalence between flatness of a vector bundle and the vanish of curvature, how is the red equal sign in the screenshot achieved? To me it seems like he is using some form of torsion free condition of the connection, i.e. $da+\omega' a$=0 . But the author did not mention torsion in this section. How did the author get the equation?
Another question I have is, does integrability of the differential equation imply the solvability of it? I don't have much knowledge on it so any reference about this would also be helpful.
If you look at the equation above the one above the one where you underlined, $da=-\omega’ a$. You replace $da$ in the left hand of the equality you underlined and you will get the right hand.