The children of Anne Marie could in theory take the throne of Denmark, because the Succession Act does not exclude anyone of the decrendant og Chr. IX and Alexandrine. Reardless what ever title they hold "to" "of" or nothing at all. Neighter does it say anything about the whole matter. It is short and simple.

§ 1. The throne is inherited down within king Christian X og Queen Alexandrines decrendant.
§ 2. When a king dies the throne goes to his son or daughter, in such way that son before daughter, if there is more of same gender oldest before youngest. If one of the children dies, that persons children takes the place after the liniar rule and the in § 1 mentioned rules.
§ 3. If a king dies without leaving direct heirs to the throne, the throne goes to his brother or sister, brother before sister. If there are more of same gender or someone have dies the rules in § 2 is to be used.
§ 4. If there are no direct heir to the throne according to § 2 and 3 the throne goes to the nearest sideline amongst the decendant of King Christian X and Queen Alexandrine according to the liniar rule, brother before sister and older before younger as set in § 2 and 3.
§ 5. Only children born in legal marriage can inheritage the throne.
The kings marriage has to have the parliaments aproval.
If one in line to the throne marries without the kings approval that person lose the right to the throne for him/her self and his/her children. .
§ 6. The rules in § 2-5 also applies if the king renounce the throne.
§ 7. This law enters into force on5. juni 1953.
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