Horst is writing what I regard as an important textbook, in which all details such as this proof can given. He asked for the proof of Eq. (16) of UFT142, a paper which gives simplified proofs of the first and second Maurer Cartan structure equation, the foundational equations of differential geometry and ECE theory. This proof was left as the proverbial "exercise for the student" by Sean Carroll, in a Harvard graduate class. Carroll gives only the proof of the first structure equation in his chapter three of "Spacetime and Geometry". All the proofs left out by Caroll are given in the UFT series.