let PTN be Petri_net; :: thesis: for S0 being Subset of the carrier of PTN
for x being object holds
( x in *' S0 iff ex f being T-S_arc of PTN ex s being place of PTN st
( s in S0 & f = [x,s] ) )
let S0 be Subset of the carrier of PTN; :: thesis: for x being object holds
( x in *' S0 iff ex f being T-S_arc of PTN ex s being place of PTN st
( s in S0 & f = [x,s] ) )
let x be object ; :: thesis: ( x in *' S0 iff ex f being T-S_arc of PTN ex s being place of PTN st
( s in S0 & f = [x,s] ) )
thus ( x in *' S0 implies ex f being T-S_arc of PTN ex s being place of PTN st
( s in S0 & f = [x,s] ) ) :: thesis: ( ex f being T-S_arc of PTN ex s being place of PTN st
( s in S0 & f = [x,s] ) implies x in *' S0 )

proof

assume x in *' S0 ; :: thesis: ex f being T-S_arc of PTN ex s being place of PTN st
( s in S0 & f = [x,s] )
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 & f = [t,s] ) by A2;
take f ; :: thesis: ex s being place of PTN st
( s in S0 & f = [x,s] )
take s ; :: thesis: ( s in S0 & f = [x,s] )
thus ( s in S0 & f = [x,s] ) by A1, A3; :: thesis: verum

end;

given f being T-S_arc of PTN, s being place of PTN such that A4: s in S0 and
A5: f = [x,s] ; :: thesis: x in *' S0
x = f `1 by A5;
hence x in *' S0 by A4, A5; :: thesis: verum