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
  (Pre t1) + (Pre t2) c= m implies
  fire(t2, fire(t1,m)) = (m - (Pre t1) - Pre t2) + (Post t1) + (Post t2)
proof
  assume
A1: (Pre t1) + (Pre t2) c= m;
A2: (Pre t1) c= (Pre t1) + (Pre t2) by Th4;
  then
A3: (Pre t1) c= m by A1,Th2;
  then
A4: fire(t1, m) = m - (Pre t1) + (Post t1) by Def7;
A5: Pre t2 = (Pre t2) + (Pre t1) - (Pre t1) by Th7;
A6: (Pre t1) + (Pre t2) - (Pre t1) c= m - (Pre t1) by A1,A2,Th8;
  m - (Pre t1) c= m - (Pre t1) + (Post t1) by Th4;
  then (Pre t2) c= fire(t1, m) by A4,A5,A6,Th2;
  hence fire(t2, fire(t1, m)) = fire(t1, m)- (Pre t2) + (Post t2) by Def7
    .= (m - (Pre t1) + (Post t1)) - (Pre t2) + (Post t2) by A3,Def7
    .= (m - (Pre t1) - (Pre t2)) + (Post t1) + (Post t2) by A1,A2,A5,Th8,Th9;
end;
