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|^m + b|^m as non zero Nat;
    reconsider d = a|^m - b|^m as non zero Integer by A0,POW1;
    a|^(2*m) = (a|^m)|^2 & b|^(2*m) = (b|^m)|^2 by NEWTON:9; then
    a|^(2*m) - b|^(2*m) = (a|^m - b|^m)*(a|^m + b|^m) by NEWTON01:1; then
    2 |-count (a|^(2*m) - b|^(2*m)) = 2 |-count c + 2 |-count d
      by NAT328,INT_2:28
    .= 2 |-count (a|^m - b|^m) + 1 by SC1;
    hence thesis;
  end;
