Congruence condition on number of subrings of given prime power order in nilpotent Lie ring

Statement
Suppose $$L$$ is a finite fact about::nilpotent Lie ring and $$p^r$$ is a prime power dividing the order of $$L$$. Then, the number of Lie subrings of $$L$$ of order $$p^r$$ is congruent to $$1$$ modulo $$p$$.

Similar facts

 * Congruence condition on number of ideals of given prime power order in nilpotent Lie ring
 * Congruence condition on number of subrings of given prime power order in nilpotent ring
 * Congruence condition on number of subgroups of given prime power order

Opposite facts

 * Congruence condition fails for number of subrings of given prime power order

Facts used

 * 1) uses::Congruence condition on number of subrings of given prime power order in nilpotent ring

Proof
The proof follows from Fact (1), which is a more general formulation (here ring means a not necessarily associative or Lie ring).