reserve PTN for Petri_net;
reserve S0 for Subset of the carrier of PTN;

theorem
  *'S0 = {f`1 where f is T-S_arc of PTN : f`2 in S0}
proof
  thus *'S0 c= {f`1 where f is T-S_arc of PTN : f`2 in S0}
  proof
    let x be object;
    assume x in *'S0;
    then consider t being transition of PTN such that
A1: x = t and
A2: ex f being T-S_arc of PTN, 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 thesis by A1,A3;
  end;
  let x be object;
  assume x in {f`1 where f is T-S_arc of PTN : f`2 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 thesis by A5;
end;
