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
  bool { x } = { {} , { x }}
proof
  now
    let y;
     reconsider yy = y as set by TARSKI:1;
    yy c= { x } iff y = {} or y = { x } by Lm3;
    hence y in bool { x } iff y in { {}, { x }} by Def1,TARSKI:def 2;
  end;
  hence thesis by TARSKI:2;
end;
