let PTN be Petri_net; :: thesis: for S0 being Subset of the carrier of PTN holds *' S0 = { (f `1) where f is T-S_arc of PTN : f `2 in S0 }
let S0 be Subset of the carrier of PTN; :: thesis: *' S0 = { (f `1) where f is T-S_arc of PTN : f `2 in S0 }
thus *' S0 c= { (f `1) where f is T-S_arc of PTN : f `2 in S0 } :: according to XBOOLE_0:def 10 :: thesis: { (f `1) where f is T-S_arc of PTN : f `2 in S0 } c= *' S0

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in *' S0 or x in { (f `1) where f is T-S_arc of PTN : f `2 in S0 } )
assume x in *' S0 ; :: thesis: x in { (f `1) where f is T-S_arc of PTN : f `2 in S0 }
then consider t being transition of PTN such that
A1: x = t and
A2: ex f being T-S_arc of PTN ex s being place of PTN st
( s in S0 & f = [t,s] ) ;
consider f being T-S_arc of PTN, s being place of PTN such that
A3: s in S0 and
A4: f = [t,s] by A2;
( f `1 = t & f `2 = s ) by A4;
hence x in { (f `1) where f is T-S_arc of PTN : f `2 in S0 } by A1, A3; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { (f `1) where f is T-S_arc of PTN : f `2 in S0 } or x in *' S0 )
assume x in { (f `1) where f is T-S_arc of PTN : f `2 in S0 } ; :: thesis: x in *' S0
then consider f being T-S_arc of PTN such that
A5: ( x = f `1 & f `2 in S0 ) ;
f = [(f `1),(f `2)] by MCART_1:21;
hence x in *' S0 by A5; :: thesis: verum