Expressing unsigned comparison through signed comparison of 2’s complement

Let n > 0 be a natural number and for any two reminders a, b modulo 2^n we have that a < b iff a xor 0x800..00 <(signed) b xor 0x800...00.

It is also true that a <(signed) b iff (0x800...00 & b <= 0x800...00 & a) && (a & 0x7FF...FFF < b & 0x7FF...FFF).

Is there a way to prove the equivalence of these two propositions?