Artificial intelligence – Determination of the consistency of the huerística and the admissible of the hueística

Given the admissible heuristics h1 and h2.

h (n) = min {h1 (n), h2 (n)}

Proving admissible
h (n) <= h1 (n) or h (n) <= h2 (n)
Therefore this is admissible

Proving consistency
h (n) <= min (h1 (n ^ prime) + c (n, a, n ^ prime), h2 (n ^ prime) + c (n, a, n ^ prime))

Am I proving to be admissible and coherent correctly?