That's a single class defined across multiple declarations, not two different classes. You only need to define the inheritance model in a single declaration, e.g.:
public class Foo { }
//Bar extends Foo
public partial class Bar : Foo { }
public partial class Bar { }
However, if you were to try the following, you'd generate a compiler error of "Partial declarations of 'Bar' must not specify different base classes":
public class Foo { }
public partial class Bar : Foo { }
public partial class Bar : object { }