To prove the first part of condition (2), we assume that the dimensioning is proper and we need to show that [Mathematical Expression Omitted]
Since proper dimensioning implies completeness, each bar [H.sub.k] is related by a dimension-set to at least one other parallel bar [H.sub.l], so [Mathematical Expression Omitted] is H, the set of all the horizontal bars.
For the second part of condition (2), we are given that [Mathematical Expression Omitted] and we need to show that the dimensioning is complete.
The normalon proper dimensioning theorem can be rephrased more compactly as the following corollary:
A normalon of rank r is dimensioned properly if and only if each of its horizontal and vertical dimensioning graphs is an r node tree.
The requirement that the graph be a tree takes care of the nonredundancy of the dimensioning, while the requirement that the tree has exactly r nodes guarantees the dimensioning's completeness.
The proper dimensioning theorem provides a sound theoretical basis for a variety of operations in the view-level analysis phase of engineering drawing understanding.
Figure 9 is an example of an improper dimensioning of the normalon of Figure 4, in which only the first condition is met.
Furthermore, if k = 1, then the dimensioning graph consists of two disconnected components, and the search can be guided by the fact that the missing dimension-set should denote the distance between a bar in one component of the graph and a bar in the other component.