
theorem Th10:
  for PTN being Petri_net, M0 being Boolean_marking of PTN, t
  being transition of PTN holds t is_firable_on M0 iff <*t*> is_firable_on M0
proof
  let PTN be Petri_net, M0 be Boolean_marking of PTN, t be transition of PTN;
  hereby
    set M = <*Firing(<*t*>/.1,M0)*>;
A1: M/.1 = Firing(<*t*>/.1,M0) by FINSEQ_4:16;
A2: now
A3:   len <*t*> = 0 + 1 by FINSEQ_1:39;
      let i be Element of NAT;
      assume i < len <*t*> & i > 0;
      hence <*t*>/.(i+1) is_firable_on M/.i & M/.(i+1) = Firing(<*t*>/.(i+1),M
      /.i) by A3,NAT_1:13;
    end;
    assume t is_firable_on M0;
    then
A4: <*t*>/.1 is_firable_on M0 by FINSEQ_4:16;
    len <*t*> = 1 by FINSEQ_1:39
      .= len M by FINSEQ_1:39;
    hence <*t*> is_firable_on M0 by A4,A1,A2;
  end;
  assume <*t*> is_firable_on M0;
  then
  ex M being FinSequence of Bool_marks_of PTN st len <*t*> = len M & <*t*>
/.1 is_firable_on M0 & M/.1 = Firing(<*t*>/.1,M0) & for i being Element of NAT
  st i < len <*t*> & i > 0 holds <*t*>/.(i +1) is_firable_on M/.i & M/.(i+1) =
  Firing(<*t*>/.(i+1),M/.i);
  hence thesis by FINSEQ_4:16;
end;
