theorem
  dom psi <> {} & dom psi is limit_ordinal & psi is increasing & A
  is_limes_of psi implies A is_cofinal_with dom psi
proof
  assume that
A1: dom psi <> {} & dom psi is limit_ordinal and
A2: psi is increasing and
A3: A is_limes_of psi;
  take psi;
  thus dom psi = dom psi;
  sup psi = lim psi & A = lim psi by A1,A2,A3,ORDINAL2:def 10,ORDINAL4:8;
  hence thesis by A2,ORDINAL2:49;
end;
