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 n be odd Nat holds 3 divides a|^n - b|^n iff 3 divides a - b
  proof
    let n be odd Nat;
    consider k such that
A0: n = 2*k + 1 by ABIAN:9;
    3 divides a|^n - b|^n implies 3 divides a - b
    proof
      assume
      A1: 3 divides a|^n - b|^n;
      3 divides a|^n - a & 3 divides -(b|^n - b)
        by A0,NEWTON02:173,INT_2:10; then
      3 divides (a|^n - a) + -(b|^n - b) by WSIERP_1:4; then
      3 divides (a|^n - b|^n) + -(a - b); then
      3 divides -(a-b) by A1,INT_2:1;
      hence thesis by INT_2:10;
    end;
    hence thesis by D3;
  end;
