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-b implies not 4 divides a|^n + b|^n
  proof
    let a,b be odd Nat;
    assume
A1: 4 divides a-b;
    per cases;
    suppose n is odd;
      hence thesis by A1,NEWTON02:69;
    end;
    suppose n is even; then
      4 divides a|^n - b|^n by NEWTON02:65;
      hence thesis by NEWTON02:58;
    end;
  end;
