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 m be non zero even Nat, a,b be odd Nat st a <> b holds
    2 |-count (a|^(2*m) - b|^(2*m)) >= 2 |-count (a|^m - b|^m) + 1
  proof
    let m be non zero even Nat, a,b be odd Nat such that
    A0: a <> b;
    reconsider c = a|^(2*m) - b|^(2*m) as non zero Integer by A0,POW1;
    reconsider d = a|^m - b|^m as non zero Integer by A0,POW1;
    A1: 2 is non trivial;
    2 |^(2 |-count d) divides d & not 2 |^((2 |-count d) +1) divides d
      by Def6; then
    2 |^((2 |-count d) +1) divides c by Even;
    hence thesis by A1,Count1;
  end;
