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
  for a,b be odd Nat holds
  (4 divides a|^n - b|^n) implies not 4 divides a|^(2*n) + b|^(2*n)
  proof
    let a,b be odd Nat;
A0: (a|^n)|^2 = a|^(2*n) & (b|^n)|^2 = b|^(2*n) by NEWTON:9;
    assume 4 divides a|^n - b|^n; then
    4 divides (a|^n + b|^n)*(a|^n - b|^n) by INT_2:2; then
    4 divides (a|^n)|^2 - (b|^n)|^2 by NEWTON01:1;
    hence thesis by NEWTON02:58,A0;
  end;
