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

theorem Th4:
  for x being object holds x in S0*' iff ex f being S-T_arc of PTN, s
  being place of PTN st s in S0 & f = [s,x]
proof
  let x be object;
  thus x in S0*' implies ex f being S-T_arc of PTN, s being place of PTN st s
  in S0 & f = [s,x]
  proof
    assume x in S0*';
    then consider t being transition of PTN such that
A1: x = t and
A2: ex f being S-T_arc of PTN, s being place of PTN st s in S0 & f = [ s,t];
    consider f being S-T_arc of PTN, s being place of PTN such that
A3: s in S0 & f = [s,t] by A2;
    take f, s;
    thus thesis by A1,A3;
  end;
  given f being S-T_arc of PTN, s being place of PTN such that
A4: s in S0 and
A5: f = [s,x];
  x = f`2 by A5;
  hence thesis by A4,A5;
end;
