To learn about other rules of writing in semi-cursive script (行書), follow this link.
This rule corresponds with the rule #6 of writing Chinese calligraphy in cursive script (草書). The concept is very similar, and the aim is to reduce the characters structure and make it more compact. Also, it speeds up writing, which is one of main purposes of semi-cursive and cursive scripts.
In case of the character 戸 (door), except the top horizontal line, which was reduced to a dot, the character structure could be classified as a relaxed standard script (楷書). The differences between the regular script and semi-cursive can vary from be subtle to drastic. Sometimes, it may be difficult to classify a single character, especially that there are many various types of standard and semi-cursive scripts.
Semi-cursive script will offer fewer opportunities for reducing lines to dots, in comparison with cursive script (草書).
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