 reserve CPN for with-nontrivial-ColoredSet Colored-PT-net;
 reserve m for colored-state of CPN;
 reserve t for Element of the carrier' of CPN;

theorem
:: Th4:
  for D being thin_cylinder of the ColoredSet of CPN, *'{t} holds
  (ex ColD being color-threshold of D st t is_firable_on m, ColD)
  iff
  D in firable_set_on(m, t)
proof
  let D be thin_cylinder of the ColoredSet of CPN, *'{t};
  thus (ex ColD being color-threshold of D st t is_firable_on m, ColD) implies
  D in firable_set_on(m, t);
  assume D in firable_set_on(m, t); then
  ex D0 be thin_cylinder of the ColoredSet of CPN, *'{t} st D=D0 &
  ex ColD being color-threshold of D0 st t is_firable_on m, ColD;
  hence thesis;
end;
