theorem Th23:
  fire <*e*> = fire e
proof
  consider F being Function-yielding FinSequence such that
A1: fire <*e*> = compose(F, Funcs(P, NAT)) and
A2: len F = len <*e*> and
A3: for i being Element of NAT st i in dom <*e*> holds
  F.i = fire (<*e*>/.i qua Element of N) by Def10;
A4: len <*e*> = 1 by FINSEQ_1:40;
  dom <*e*> = {1} by FINSEQ_1:2,def 8;
  then
A6: 1 in dom <*e*> by TARSKI:def 1;
  then
A7: <*e*>/.1 = <*e*>.1 by PARTFUN1:def 6;
  F.1 = fire (<*e*>/.1 qua Element of N) by A3,A6;
  then
A8: F = <*fire e*> by A2,A4,A7,FINSEQ_1:40;
  dom fire e c= Funcs(P,NAT) by Def8;
  hence thesis by A1,A8,FUNCT_7:46;
end;
