reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  a1|^(n+k) + b1|^(n+k) = a1|^n*(a1|^k + b1|^k) + b1|^k*(b1|^n-a1|^n)
  proof
    a1|^(n+k) = a1|^n * a1|^k & b1|^(n+k) = b1|^n*b1|^k by NEWTON:8;
    hence thesis;
  end;
