The existence of an observer O may be considered to be the origin of a coordinate plane representing the emotional and intellectual awareness of the observer. The measure of the development of the observer may be represented by a measure r as the radius of a circle drawn about O, with the area R bounded by this circle representing the emotional and intellectual awareness of the observer. For this reason, the plane may be called the plane of awareness.

The relationship of O to another entity O' may be represented by locating O' at a point on the plane of awareness, with the length of the line d drawn between O and O' representing the perceived distance between the entities. The awareness of O' is similarly represented by a circle drawn about O' of radius r' and area R' likewise representing development and awareness respectively.

In any relationship of O and O', the circles may intersect at no points, at a single point, or at two points. The case of congruent circles overlapping at infinitely many points can happen only when both O and O' represent the same individual, which is considered a degenerate case, the discussion of which is beyond the scope of this exercise.

In situations where there is intersection between O and O', a lens shaped area C is formed by the overlapping area bounded by the intersecting perimeters of O and O'. The measure of this area represents the emotional and intellectual bond between O and O', and will vary with the distance d between individuals, as well as the relative radii r and r' and areas R and R'.

In exceptional circumstances, O and O' may overlap sufficiently that their centers are mutually located within the area C, in which case the entities may be described as having particular closeness.

At first glance, it may appear that as the distance d between O and O' increases at a rate dd, the area C must necessarily diminish, representing a diminishing bond and loss of closeness between the entities. This view however does not take into consideration the continuing development of O and O'.

As O and O' continue to develop in awareness, the measures r and r' increase respectively at rates dr and dr', with corresponding increases in areas dR and dR'. It can be demonstrated that as long as the relationship of dd, dr and dr' satisfies the inequality dd <= (dr + dr'), there will be no loss in closeness C between entities.

It may also be demonstrated that as long as the above inequality is satisfied, that the area C will continue to increase at a rate dC = (dr + dr' - dd) * (dr + dr' - dd), proving that the bond between entities will be not only preserved but strengthened as long as both individuals continue to grow.

Q.E.D.