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 Th20:
  t*z divides (t+z)|^n - (t-z)|^n + ((-z)|^n - z|^n)
  proof
 A1:(t+z)|^n - (t|^n + z|^n) =
    (t+(-z))|^n - (t|^n + (-z)|^n) + ((t+z)|^n -(t-z)|^n + ((-z)|^n - z|^n));
    per cases;
    suppose n = 0; then
      1*(t+z)|^n - 1*(t-z)|^n + (1*(-z)|^n - 1*z|^n) = 0;
      hence thesis by INT_2:12;
    end;
    suppose n>0; then
      t*z divides (t-z)|^n - (t|^n + (-z)|^n) & t*z divides
        (t+z)|^n - (t|^n + z|^n) by Th17,Th19;
      hence thesis by A1,INT_2:1;
    end;
  end;
