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 st |.a.| <> |.b.| holds
    2 |-count (a-b)|^2 <> 2|-count (a+b)|^2 &
    2 |-count (a-b)|^2 <> 2|-count a|^2-b|^2
    proof
      let a,b be odd Integer such that
  A1: |.a.| <> |.b.|;
 A1a: a - b <> a - a by A1;
      reconsider c = a-b as non zero Integer by A1;
      reconsider d = a+b as non zero Integer by A1,ABS1;
  A2: 2 |-count (a-b)|^2 = 2 |-count c|^2
      .= 2*(2 |-count (a-b)) by NAT332,INT_2:28;
  A3: 2 |-count (a+b)|^2 = 2 |-count d|^2
      .= 2*(2 |-count (a+b)) by NAT332,INT_2:28;
      2*2 = 2|^2 by NEWTON:81; then
  A6: not (2|^2 divides (a-b) iff 2|^2 divides (a+b)) by NEWTON0258;
      a+b <> 0 & 2 is non trivial by A1,ABS1;
      hence thesis by A2,A3,A6,A1a,DIC;
    end;
