Theorem:

Consider Atoms A, B, C where:
{A}:::{B} = 1,
{B}:::{C} = 1,
{A} does not equal {C},
and A, B, and C are all in the same Domain.

If {A}:::{B}, then A and B must each have two essential letters, and may not share any between themselves.

Thus, there are only 3! = 6 possible atoms that A can bond with, and B is one of them. The situation is the same for B and C, respectively.

Consider then A and C. A has occupied two of the Essential Letters, and B has occupied an additional two discreet from A’s. There is only one Letter left in the set of five; thus, C must take an Essential Letter already in use by A (not B because it has been previously stated that {B}:::{C}), and therefore {A}:::{C} = 0.

A more explicit form of the conclusion: {A}:::{B}&&{B}:::{C}&&{C}:::{A} = 0