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

theorem Th15:
  S.:*' = *'S
proof
  thus S.:*' c= *'S
  proof
    let x be object;
    assume x in S.:*';
    then consider f being S-T_arc of PTN.:, s being place of PTN.: such that
A1: s in S.: and
A2: f = [s,x] by Th4;
    [x,.:s] is T-S_arc of PTN by A2,RELAT_1:def 7;
    hence thesis by A1,Th2;
  end;
  let x be object;
  assume x in *'S;
  then consider f being T-S_arc of PTN, s being place of PTN such that
A3: s in S and
A4: f = [x,s] by Th2;
  [s.:,x] is S-T_arc of PTN.: by A4,RELAT_1:def 7;
  hence thesis by A3,Th4;
end;
