theorem
  A is_cofinal_with 0 implies A = 0
proof
  given psi such that
A1: dom psi = 0 and
  rng psi c= A and
  psi is increasing and
A2: A = sup psi;
  thus A = sup 0 by A1,A2,RELAT_1:42
    .= 0 by Th18;
end;
