reserve x, y for set;

theorem Th11:
  derangements {} = {{}}
  proof
    hereby let x be object;
      assume x in derangements {}; then
      ex f be Permutation of {} st x = f & f is without_fixpoints;
      hence x in {{}} by FUNCT_2:9,127;
    end;
    let x be object;
    assume x in {{}}; then
A1: x = {} by TARSKI:def 1;
    rng (id {}) = {};
    then id {} is Permutation of {} by FUNCT_2:57;
    hence thesis by A1,Th1;
  end;
