Connection difference as an endomorphism-valued one-form ($D_1-D\in A^1(\mathrm{End}E)$)

The graded Leibniz rule terms of two connections cancel, making their difference $C^{\infty }$-linear in the section. It is therefore an element of $A^1(\mathrm{End}E)$. Conversely adding any such form to a connection gives another connection, making the space of connections an affine space.