let PTN be Petri_net; :: thesis: ({} the carrier of PTN) *' = {}
set x = the Element of ({} the carrier of PTN) *' ;
assume not ({} the carrier of PTN) *' = {} ; :: thesis: contradiction
then the Element of ({} the carrier of PTN) *' in ({} the carrier of PTN) *' ;
then ex t being transition of PTN st
( the Element of ({} the carrier of PTN) *' = t & ex f being S-T_arc of PTN ex s being place of PTN st
( s in {} the carrier of PTN & f = [s,t] ) ) ;
hence contradiction ; :: thesis: verum