reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem
  for X,Y,Z being set st X in U & Y in U & Z in U holds
  bool [: [:bool [:X, Y:],bool [:X,Y:]:],bool [:Z,Y:] :] in U
  proof
    let X,Y,Z be set;
    assume that
A1: X in U and
A2: Y in U and
A3: Z in U;
    reconsider X,Y,Z as Element of U by A1,A2,A3;
    bool [: [:bool [:X, Y:],bool [:X,Y:]:],bool [:Z,Y:] :] in U;
    hence thesis;
end;
