reserve x, y for set;

theorem Th1:
  id {} is without_fixpoints
  proof
    assume id {} is with_fixpoint; then
    consider y being object such that
A1: y is_a_fixpoint_of (id {}) by ABIAN:def 5;
    y in dom (id {}) by A1,ABIAN:def 3;
    hence thesis;
  end;
