reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem :: ZFMISC_1:30
  bool (I --> {x}) = I --> { {} , {x} }
proof
  now
    let i be object;
    assume
A1: i in I;
    hence (bool (I --> {x})).i = bool ((I --> {x}).i) by Def1
      .= bool {x} by A1,FUNCOP_1:7
      .= { {} , {x} } by ZFMISC_1:24
      .= (I --> { {} , {x} }).i by A1,FUNCOP_1:7;
  end;
  hence thesis;
end;
