let PTN be Petri_net; :: thesis: for S0 being Subset of the carrier of PTN holds S0 *' = { (f `2) where f is S-T_arc of PTN : f `1 in S0 }
let S0 be Subset of the carrier of PTN; :: thesis: S0 *' = { (f `2) where f is S-T_arc of PTN : f `1 in S0 }
thus S0 *' c= { (f `2) where f is S-T_arc of PTN : f `1 in S0 } :: according to XBOOLE_0:def 10 :: thesis: { (f `2) where f is S-T_arc of PTN : f `1 in S0 } c= S0 *'

proof

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

end;

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