theorem
  fire(t2, fire(t1,m)) = ((fire t2)*(fire t1)).m
proof
  dom fire t1 = Funcs(P, NAT) by Def8;
  then
A1: m in dom(fire t1) by FUNCT_2:8;
  thus fire(t2, fire(t1,m)) = (fire t2).fire(t1,m) by Def8
    .= (fire t2).((fire t1).m) by Def8
    .= ((fire t2)*(fire t1)).m by A1,FUNCT_1:13;
end;
