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+u-z)|^2,(t|^2+u|^2+z|^2) are_congruent_mod 2
  proof
    (t-z+u)|^2=t|^2+z|^2+u|^2-2*t*z+2*t*u-2*z*u by SERIES_4:3; then
    (t-z+u)|^2 - (t|^2+z|^2+u|^2) = 2*(t*u-t*z-z*u);
    hence thesis;
  end;
