reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;

theorem
  W is Tarski iff W is subset-closed & (for X st X in W holds bool X in W) &
  for X st X c= W & card X in card W holds X in W
proof
  hereby
    assume
A1: W is Tarski;
    hence
W is subset-closed & for X st X in W holds bool X in W;
    let X;
    assume that
A2: X c= W and
A3: card X in card W;
 card X <> card W by A3;
then  not X,W are_equipotent by CARD_1:5;
    hence X in W by A1,A2;
  end;
 now
    assume
A4: for X st X c= W & card X in card W holds X in W;
    let X;
    assume X c= W;
then  card X c= card W & not card X in card W or X in W by A4,CARD_1:11;
    hence X,W are_equipotent or X in W by CARD_1:3,5;
  end;
  hence thesis;
end;
