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

theorem
  (c1|^n/2+c1/2)|^2-(c1|^n/2-c1/2)|^2 = c1|^(n+1)
  proof
    set k = c1|^n/2+c1/2;
    set l = c1|^n/2-c1/2;
    k|^2 - l|^2 = k*k -l|^2 by NEWTON:81
    .= k*k -l*l by NEWTON:81
    .= c1*c1|^n
    .= c1|^(n+1) by NEWTON:6;
    hence thesis;
  end;
