
theorem Th5:
  for PTN being Petri_net, M0 being Boolean_marking of PTN, t
being transition of PTN, s being place of PTN st s in {t}*' holds Firing(t,M0).
  s = TRUE
proof
  let PTN be Petri_net, M0 be Boolean_marking of PTN, t be transition of PTN,
  s be place of PTN;
  set M = M0 +* (*'{t}-->FALSE) +* ({t}*'-->TRUE);
A1: [#]the carrier of PTN = the carrier of PTN;
A2: dom M0 = the carrier of PTN & dom (*'{t}-->FALSE) = (*'{t})
 by FUNCT_2:def 1;
  dom ({t}*'-->TRUE) = ({t}*') by FUNCT_2:def 1;
  then
A3: dom M = dom(M0 +* (*'{t}-->FALSE)) \/ {t}*' by FUNCT_4:def 1
    .= (the carrier of PTN) \/ (*'{t}) \/ ({t}*') by A2,FUNCT_4:def 1
    .= (the carrier of PTN) \/ ((*'{t}) \/ ({t}*')) by XBOOLE_1:4
    .= the carrier of PTN by A1,SUBSET_1:11;
  assume
A4: s in {t}*';
  then ((M0 +* (*'{t}-->FALSE)) +* ({t}*'-->TRUE)).:({t}*') = {TRUE} by Th4;
  then M.s in {TRUE} by A4,A3,FUNCT_1:def 6;
  hence thesis by TARSKI:def 1;
end;
