reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th65:
  for a,b be odd Nat, m be even Nat holds 4 divides a|^(m) - b|^(m)
  proof
    let a,b be odd Nat, m be even Nat;
    consider n such that
    L0: m = 2*n by ABIAN:def 2;
    L1: 4 divides a + b implies 4 divides a|^(m)-b|^(m)
    proof
      assume
      A1: 4 divides a + b;
      a+b divides a|^(m)-b|^(m) by NEWTON01:36,L0;
      hence thesis by A1,INT_2:9;
    end;
    4 divides a - b implies 4 divides a|^(m)-b|^(m) by Th64;
    hence thesis by L1,Th58;
  end;
