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 Integer holds 2 |-count (a|^2 + b|^2) = 1
  proof
    let a,b be odd Integer;
    reconsider t = a|^2, u = b|^2 as odd Nat;
A2: 2*2 = 2|^2 by NEWTON:81;
    2 divides a - b & 2 divides a + b by ABIAN:def 1; then
    2*2 divides (a-b)*(a+b) by NAT31; then
    4 divides (a|^2 - b|^2) by NEWTON01:1; then
    2|^1 divides (a|^2 + b|^2) & not 2|^(1+1) divides (a|^2 + b|^2)
      by ABIAN:def 1,A2,NEWTON0258;
    hence thesis by NAT_3:def 7;
  end;
