let PTN be PT_net_Str ; :: thesis: for S being Subset of holds
( S is Trap-like iff S .: is Deadlock-like )
let S be Subset of ; :: thesis: ( S is Trap-like iff S .: is Deadlock-like )
A1: (S .: ) *' = *' S by Th15;
thus ( S is Trap-like implies S .: is Deadlock-like ) :: thesis: ( S .: is Deadlock-like implies S is Trap-like )

proof

assume S *' is Subset of ; :: according to PETRI:def 7 :: thesis: S .: is Deadlock-like
hence *' (S .: ) is Subset of by A1, Th16; :: according to PETRI:def 5 :: thesis: verum

end;

assume *' (S .: ) is Subset of ; :: according to PETRI:def 5 :: thesis: S is Trap-like
hence S *' is Subset of by A1, Th16; :: according to PETRI:def 7 :: thesis: verum