Transiter master theorem for composition operator on subgroup properties

Statement
This is the version of the transiter master theorem for the fact about::composition operator on subgroup properties.

The version for left transiters
Suppose $$p$$ is a fact about::subgroup property. Let $$L(p)$$ denote the left transiter of $$p$$.


 * 1) If $$p$$ is transitive, $$p$$ is stronger than its left transiter. That is, $$p \le L(p)$$.
 * 2) If $$p$$ is identity-true, the left transiter of $$p$$ is stronger than $$p$$. That is, $$L(p) \le p$$.
 * 3) if $$p$$ is a t.i. subgroup property, i.e., $$p$$ is both transitive and identity-true, then $$p = L(p)$$.
 * 4) The left transiter of any property is a t.i. subgroup property, that is, it is both transitive and identity-true.

Together, (3) and (4) tell us that the left transiter operator is idempotent: $$L(L(p)) = L(p)$$

The version for right transiters
Suppose $$p$$ is a subgroup property. Let $$R(p)$$ denote the right transiter of $$p$$.


 * 1) If $$p$$ is transitive, $$p$$ is stronger than its right transiter. That is, $$p$$ \le R(p).
 * 2) If $$p$$ is identity-true, the right transiter of $$p$$ is stronger than $$p$$. That is, $$R(p) \le p$$.
 * 3) if $$p$$ is a t.i. subgroup property, i.e., $$p$$ is both transitive and identity-true, then $$p = R(p)$$.
 * 4) The right transiter of any property is a t.i. subgroup property, that is, it is both transitive and identity-true.

Together, (3) and (4) tell us that the left transiter operator is idempotent: $$L(L(p)) = L(p)$$. Moreover, the fixed point space of this operator is the same as the fixed point space of the left transiter.