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 non zero Integer, p be odd prime Nat st
    |.a.| <> |.b.| & not p divides b holds
    p |-count (a|^2 - b|^2) = max (p |-count (a-b), p |-count (a+b))
  proof
    let a,b be non zero Integer, p be odd prime Nat such that
A1: |.a.| <> |.b.| & not p divides b;
A2: p is non trivial & a-b <> 0 & a+b <> 0 by A1,ABS1;
    p divides (a-b) implies not p divides (a+b) by A1,SUD; then
A3: p |-count (a-b) <> 0 implies p |-count (a+b) = 0 by A2,NAT327;
    p |-count (a|^2 - b|^2) = p |-count ((a-b)*(a+b)) by NEWTON01:1
    .= (p |-count (a-b)) + (p |-count (a+b)) by A2,NAT328;
    hence thesis by A3,XXREAL_0:def 10;
  end;
