theorem Th81:
  n is odd implies n,1 are_congruent_mod 4 or n,3 are_congruent_mod 4
  proof
    assume n is odd;
    then n mod 4 = 1 or n mod 4 = 3 by Th80;
    then n mod 4 = 1 mod 4 or n mod 4 = 3 mod 4;
    hence thesis by NAT_D:64;
  end;
