theorem
  fire <*e1,e2*> = (fire e2)*(fire e1)
proof
  consider F being Function-yielding FinSequence such that
A1: fire <*e1,e2*> = compose(F, Funcs(P, NAT)) and
A2: len F = len <*e1,e2*> and
A3: for i being Element of NAT st i in dom <*e1,e2*> holds
  F.i = fire (<*e1,e2*>/.i qua Element of N) by Def10;
A4: len <*e1,e2*> = 2 by FINSEQ_1:44;
A7: dom <*e1,e2*> = {1,2} by A4,FINSEQ_1:2,def 3;
A8: 1 in {1,2} by TARSKI:def 2;
A9: 2 in {1,2} by TARSKI:def 2;
A10: <*e1,e2*>/.1 = <*e1,e2*>.1 by A7,A8,PARTFUN1:def 6;
A11: <*e1,e2*>/.2 = <*e1,e2*>.2 by A7,A9,PARTFUN1:def 6;
A12: F.1 = fire e1 by A3,A7,A8,A10;
  F.2 = fire e2 by A3,A7,A9,A11;
  then
A13: F = <*fire e1,fire e2*> by A2,A4,A12,FINSEQ_1:44;
  (fire e1)*id Funcs(P, NAT) = fire e1 by Th24;
  then (fire e2)*(fire e1)*id Funcs(P, NAT) = (fire e2)*(fire e1)
  by RELAT_1:36;
  hence thesis by A1,A13,FUNCT_7:51;
end;
