reserve PTN for Petri_net;
reserve S0 for Subset of the carrier of PTN;
reserve T0 for Subset of the carrier' of PTN;

theorem
  *'T0 = {f`1 where f is S-T_arc of PTN : f`2 in T0}
proof
  thus *'T0 c= {f`1 where f is S-T_arc of PTN : f`2 in T0}
  proof
    let x be object;
    assume x in *'T0;
    then consider s being place of PTN such that
A1: x = s and
A2: ex f being S-T_arc of PTN, t being transition of PTN st t in T0 &
    f = [s,t];
    consider f being S-T_arc of PTN, t being transition of PTN such that
A3: t in T0 and
A4: f = [s,t] by A2;
    f`1 = s & f`2 = t by A4;
    hence thesis by A1,A3;
  end;
  let x be object;
  assume x in {f`1 where f is S-T_arc of PTN : f`2 in T0};
  then consider f being S-T_arc of PTN such that
A5: x = f`1 & f`2 in T0;
  f = [f`1,f`2] by MCART_1:21;
  hence thesis by A5;
end;
