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He defined in  the operations union and com-
position over PN and some relations based on the func-
tioning and the result of the work of the PN. Assertions
about the connection between these operations and relations have also
been stated. In this paper we prove some of them, in particular those
which deal with PN that can be compared by the result of their work.
We shall first present the basic de definitions which are relevant to this work. For more information see . We agree on the following notations:
Everywhere below Ei is a PN of the kind:...
Union of E1 and E2 is a PN of the kind:.....
Z is a transition and Z0 is the empty transition. For the definitions
of the operations union, intersection and composition over T see .
The following relations over PN are central for this paper:...
Let us define:...
We shall assume that if a place takes part simultaneously in two or more PNs then it has one and the same notation.
Let us take arbitrary r
and x.For every T there exists W such that p
receives one and the same characteristic in both T and W.We now fix the transitions Z1 and Z2
and construct a new transition, namely Z3.
The last is true because the token
receives one and the
same characteristic in both E1 and E2, since P=Q.
We shall consider that L3 \ L2 6= ; because otherwise the operation composition over PNs is reduced to union. By the defi nition of the operation union over T and the condition E1(; x) = E2(; x), we have that the token receives one and the same characteristic in both transitions.