reserve i,j,k,l for Nat,
  x,x1,x2,y1,y2 for set;
reserve P,p,x,y,x1,x2 for set,
  m1,m2,m3,m4,m for marking of P,
  i,j,j1,j2,k,k1,k2,l,l1 for Nat;
reserve t,t1,t2 for transition of P;

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;
