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
  for a,b,m be odd Nat holds 4 divides a-b iff not 4 divides a|^m + b|^m
  proof
    let a,b,m be odd Nat;
    thus 4 divides a-b implies not 4 divides a|^m + b|^m
    proof
      assume 4 divides a-b; then
      not 4 divides a + b by Th58;
      hence thesis by Th68;
    end;
    assume not 4 divides a|^m + b|^m; then
    not 4 divides a + b by Th68;
    hence thesis by Th58;
  end;
