
theorem
::Th3:
  for I be non trivial finite set,
  CPNT be Colored-PT-net-Family of I,
  N be Function of I, bool rng CPNT,
  O be connecting-mapping of CPNT st CPNT is one-to-one &
  N, O is_Cell-Petri-nets holds
  for i be Element of I holds not CPNT.i in N.i
proof
  let I be non trivial finite set,
  CPNT be Colored-PT-net-Family of I,
  N be Function of I, bool rng CPNT,
  O be connecting-mapping of CPNT;
  assume
A1: CPNT is one-to-one;
  assume
A2: N, O is_Cell-Petri-nets;
  let i be Element of I;
  assume
A3: CPNT.i in N.i;
  N.i = {CPNT.j where j is Element of I:j <> i &
  ex t be transition of CPNT.i,
  s be object st [t, s] in O.[i, j]} by A2; then
  consider j be Element of I such that
A4: CPNT.i = CPNT.j & j <> i &
  ex t be transition of CPNT.i,
  s be object st [t, s] in O.[i, j] by A3;
  dom CPNT = I by PARTFUN1:def 2;
  hence contradiction by A1, A4, FUNCT_1:def 4;
end;
