reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th24:
  for X be denumerable set holds
  Frechet_Filter (X) = the set of all X\A where A is finite Subset of X
  proof
    let X be denumerable set;
A1: card X = omega by CARD_3:def 15;
    hereby
      let x be object such that
A2:   x in Frechet_Filter(X);
      reconsider x0=x as Subset of X by A2;
      card(X \ x0) in card X by A2,CARD_FIL:18;
      then
A3:   X \ x0 is finite Subset of X by A1;
      X \ ( X \x0 )= X /\ x0 & X /\x0 c= X by XBOOLE_1:48;
      then X \ (X \ x0)=x0 by XBOOLE_1:28;
      hence x in the set of all X\A where A is finite Subset of X by A3;
    end;
    let x be object such that
A4: x in the set of all X\A where A is finite Subset of X;
    consider a0 be finite Subset of X such that
A5: x=X \ a0 by A4;
    reconsider x0=x as Subset of X by A5;
    X\(X\a0) = X/\a0 & X/\a0 c= X by XBOOLE_1:48;
    then card (X \ x0) = card a0 by A5,XBOOLE_1:28;
    hence thesis by A1;
  end;
