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`2 where f is T-S_arc of PTN : f`1 in T0}
proof
  thus T0*' c= {f`2 where f is T-S_arc of PTN : f`1 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 T-S_arc of PTN, t being transition of PTN st t in T0 &
    f = [t,s];
    consider f being T-S_arc of PTN, t being transition of PTN such that
A3: t in T0 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`2 where f is T-S_arc of PTN : f`1 in T0};
  then consider f being T-S_arc of PTN such that
A5: x = f`2 & f`1 in T0;
  f = [f`1,f`2] by MCART_1:21;
  hence thesis by A5;
end;
