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

theorem Th8:
  for x being set holds x in T0*' iff ex f being T-S_arc of PTN, t
  being transition of PTN st t in T0 & f = [t,x]
proof
  let x be set;
  thus x in T0*' implies ex f being T-S_arc of PTN, t being transition of PTN
  st t in T0 & f = [t,x]
  proof
    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 & f = [t,s] by A2;
    take f, t;
    thus thesis by A1,A3;
  end;
  given f being T-S_arc of PTN, t being transition of PTN such that
A4: t in T0 and
A5: f = [t,x];
  x = f`2 by A5;
  hence thesis by A4,A5;
end;
