  Our first main result gives assumptions guaranteeing that proper holomorphic
maps between Cartan type I bounded symmetric domains have simple block matrix
shape, answering positively a question of Mok. The proof is based on the second
main result establishing similar phenomenon for local CR maps between arbitrary
boundary components of two bounded symmetric domains of the above type. Since
boundary components other than Shilov boundaries are Levi-degenerate, our
analysis is based on their $2$-nondegeneracy combining Levi forms with higher
order tensors.
