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

theorem
for x being set holds
  bool (I --> x) = I --> bool x
proof let x be set;
  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
      .= (I --> bool x).i by A1,FUNCOP_1:7;
  end;
  hence thesis;
end;
