reserve x, y, z, E, E1, E2, E3 for set,
  sE for Subset-Family of E,
  f for Function of E, E,
  k, l, m, n for Nat;

theorem
  for E being set, f being Function of E, E, sE being non empty covering
  Subset-Family of E st for X being Element of sE holds X misses f.:X holds
  f is without_fixpoints
proof
  let E be set, f be Function of E, E, sE be non empty covering Subset-Family
  of E;
  assume
A1: for X being Element of sE holds X misses f.:X;
  given x being object such that
A2: x is_a_fixpoint_of f;
A3: f.x = x by A2;
A4: x in dom f by A2;
  dom f = E by FUNCT_2:52;
  then x in union sE by A4,Th4;
  then consider X being set such that
A5: x in X and
A6: X in sE by TARSKI:def 4;
  f.x in f.:X by A4,A5,FUNCT_1:def 6;
  then X meets f.:X by A3,A5,XBOOLE_0:3;
  hence contradiction by A1,A6;
end;
