Proof
ex_ind
(fun x (H : 0 # x = 1)
=> distinct_0_1 (H || a = 1 @a by <- prod_0_l x)).

(** Proves that 0 does not have a multiplicative inverse - I.E. 0 does not have a reciprocal. *)
Theorem prod_0_inv
:  ~ prod_has_inv 0.