reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem
  ex M st N in M & (for X,Y holds X in M & Y c= X implies Y in M) & (for
  X holds X in M implies bool X in M) & for X holds X c= M implies X,M
  are_equipotent or X in M
proof
  consider M such that
A1: N in M and
A2: for X,Y holds X in M & Y c= X implies Y in M and
A3: for X st X in M ex Z st Z in M & for Y st Y c= X holds Y in Z and
A4: for X holds X c= M implies X,M are_equipotent or X in M by TARSKI_A:1;
  take M;
  thus N in M by A1;
  thus for X,Y holds X in M & Y c= X implies Y in M by A2;
  thus for X holds X in M implies bool X in M
  proof
    let X;
    assume X in M;
    then consider Z such that
A5: Z in M and
A6: for Y st Y c= X holds Y in Z by A3;
    now
      let Y be object;
   reconsider YY=Y as set by TARSKI:1;
      assume Y in bool X;
      then YY c= X by Def1;
      hence Y in Z by A6;
    end;
    hence thesis by A2,A5,TARSKI:def 3;
  end;
  thus thesis by A4;
end;
