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

theorem Th6:
  for x being set holds x in *'T0 iff ex f being S-T_arc of PTN, t
  being transition of PTN st t in T0 & f = [x,t]
proof
  let x be set;
  thus x in *'T0 implies ex f being S-T_arc of PTN, t being transition of PTN
  st t in T0 & f = [x,t]
  proof
    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 & f = [s,t] by A2;
    take f, t;
    thus thesis by A1,A3;
  end;
  given f being S-T_arc of PTN, t being transition of PTN such that
A4: t in T0 and
A5: f = [x,t];
  x = f`1 by A5;
  hence thesis by A4,A5;
end;
