A network of constraints T on M99 implemented by means of Lemma is denoted hence forth by Ψ g T
We can derive a contradiction from a network of constraints iff the network derives two relations R 1 and R 2 between one pair of vertices such that R 1 R 2
∅ The ways in which a contradiction can be derived in networks labelled by relations of M99  based on the intersections of relations  except the empty relation which obviously generates a contradic tion by itself  are

| CGPP   |               |                   |                |      |
|--------|---------------|-------------------|----------------|------|
| a      | CG            | e                 |                |      |
| CG     | CNO           |                   |                |      |
| b      |               |                   |                |      |
| CG     | {CGPP, CNO}   |                   |                |      |
| c      | e             |                   |                |      |
| d      | CG            | e                 | {CGPP, CGPP-1, | CNO} |
| CGPP   |               |                   |                |      |
| e      | C GPP         |                   |                |      |
| f      | CGPP          | CNO               |                |      |
| e      |               |                   |                |      |
| g      | C GPP         | {CG, CGPP-11      |                |      |
| e      |               |                   |                |      |
| C GPP  |               |                   |                |      |
| h      | e             | {CG, CNO}         |                |      |
| CGPP   | {CGPP-1' CNO} |                   |                |      |
| i      | e             |                   |                |      |
| l      | C GPP         | {CG, CGPP-1, CNO} |                |      |
| e      |               |                   |                |      |
| m      | CNO           | e                 | {CG, CGPP}     |      |
| n      | {CG, CGPP}    | e                 | {CGPP-1' CNO}  |      |

This is simply obtained by considering all pairs of relations in M 99 whose intersection is empty In G99 the contradictions are only
{CG,  CGPP} ⊕ {CG,  CGPP} ~ ⊕ {CGPP,  CGPP-1,  CNO} and
{CG, CGPP} ⊕ {CG, CNO} ⊕ {CGPP, CGPP-1, CNO}
Henceforth we represent the relation {CG, CGPP} by Z, the relation {CG, CNO} by N and the relation {CGPP, CGPP-1, CNO} by 7  Since any path in which labels are all w ccorresponds to the representation of y between each pair of vertices in the path and y న the possible expressions for contradictions in G 99 are γ n & γ n  ⊕ δ and γ n
® ✔ ¬ ™ ® ½ ® ✔ ¯ 
w" ® -("-") ® " ® "-" is called a quasi " cycle  The graph representation of these two different contradictory situations in G 99 is showed in Figure     A y cycle and a quasi 층 cycle  in M 99 force the elements involved in the cycle to be all in the relation CG
We can now show that the contradictions represented in G 99 are the only contradictions which can be obtained by the implementation suggested in Lemma      This is very important because we may perform consistency checking by simply checking all cycles  This result is stated in the next lemma Lemma 6  Given a network of constraints T on M99, T is inconsistent iff Ψ99 T   contains either a Z-cycle of a quasi Z-cycle, and two vertices of one cycle are connected by an edge labelled by 4.