● Let us analyse 'the door of Ḥāmid's house':
○ Door (بَابُ): This is the first noun. Since it is a muḍāf, it has no al- and takes a light ending, a single ḍammah: بَابُ.
○ House (بَيْتِ): This middle noun is the muḍāf ilayh of the first noun, so it is genitive with a kasrah. It is also the muḍāf of the next noun, so it cannot take al- and has a light ending: بَيْتِ.
○ Ḥāmid (حَامِدٍ): This final noun is the final muḍāf ilayh, so it remains genitive: حَامِدٍ.
● The completed phrase is بَابُ بَيْتِ حَامِدٍ.
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Detailed Analysis of the Grammatical Structure
● Let us analyse 'the door of Ḥāmid's house':
○ Door (بَابُ): This is the first noun. Since it is a muḍāf, it has no al- and takes a light ending, a single ḍammah: بَابُ.
○ House (بَيْتِ): This middle noun is the muḍāf ilayh of the first noun, so it is genitive with a kasrah. It is also the muḍāf of the next noun, so it cannot take al- and has a light ending: بَيْتِ.
○ Ḥāmid (حَامِدٍ): This final noun is the final muḍāf ilayh, so it remains genitive: حَامِدٍ.
● The completed phrase is بَابُ بَيْتِ حَامِدٍ.