reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem
  -t mod a = (u*a - (t mod a)) mod a
  proof
    thus (u*a - (t mod a)) mod a =
      ((((0+u*a)mod a) -((t mod a)mod a))mod a) by INT_6:7
    .= ((0 mod a)-(t mod a))mod a
    .= (0-t) mod a by INT_6:7
    .= -t mod a;
  end;
