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
  for PTN being Petri_net, t being transition of PTN, S0 being Subset
  of the carrier of PTN holds t in *'S0 iff {t}*' meets S0
proof
  let PTN be Petri_net;
  let t be transition of PTN;
  let S0 be Subset of the carrier of PTN;
  thus t in *'S0 implies {t}*' meets S0
  proof
    assume t in *'S0;
    then consider f being T-S_arc of PTN, s being place of PTN such that
A1: s in S0 and
A2: f = [t,s] by Th2;
    t in {t} by TARSKI:def 1;
    then s in {t}*' by A2;
    hence {t}*' /\ S0 <> {} by A1,XBOOLE_0:def 4;
  end;
  assume {t}*' /\ S0 <> {};
  then consider s being place of PTN such that
A3: s in {t}*' /\ S0 by SUBSET_1:4;
A4: s in S0 by A3,XBOOLE_0:def 4;
  s in {t}*' by A3,XBOOLE_0:def 4;
  then consider f being T-S_arc of PTN, t0 being transition of PTN such that
A5: t0 in {t} and
A6: f = [t0,s] by Th8;
  t0 = t by A5,TARSKI:def 1;
  hence thesis by A4,A6;
end;
