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Tadmeen (تضمين, literally "insertion") is a grammatical allowance introduced by Islamic scholars which attempts to explain why the Qur'an sometimes uses incorrect prepositions. The explanation by way of tadmeen is that if a word is used with an incorrect preposition, then it is because a different word which can be used with that preposition, was invisibly inserted into the original word.
In English one can say "believe in god" and "pray to god". If the Qur'an was in English and contained "believe to god", it would be an error, because it should be "believe in god". On this analogy, the way tadmeen would be applied would be to say that the word "pray" was invisibly inserted into the word "believe" and since "pray" can be used as "pray to", then "believe to" is also correct.
Translations of the Qur'an often gloss prepositional errors found in the Arabic original by hiding them in the English renditions. Arab-speaking people, if they notice these errors, generally blame their own judgement rather than the Qur'an. As a result, it is generally only translators and grammarians who are left to deal with these issues.
"They humble themselves to their lord (فأخبتوا إلى ربهم)
The verb أخبت (to humble/submit oneself) is used with the preposition ل. For example:
And so those who were given knowledge may know that it is the truth from your Lord and [therefore] believe in it, and their hearts humbly submit (فتخبت) to it (له, la-hu). And indeed is Allah the Guide of those who have believed to a straight path.
However in 11:23 it's used incorrectly with the preposition إلى:
Indeed, they who have believed and done righteous deeds and humbled themselves to their Lord (فأخبتوا إلى ربهم) - those are the companions of Paradise; they will abide eternally therein.
And the explanation given is:
وفي الآية ضُمِّن الفعل (أخبتوا) معنى (أنابوا)، فعدِّي بحرف الجر (إلى)
And in the verse the verb "humble themselves" included "return" so it was used with preposition إلى
So a word which can use that preposition was somehow inserted into it.
"Allah's servants drink with a spring" (عينا يشرب بها عباد الله)
A spring wherefrom (بها, bihaa) the slaves of Allah will drink, causing it to gush forth abundantly.
The word bihaa means "with it". It's strange in Arabic to say "drink with a spring". The correct preposition is "from" (من), "drink from it". Ibn Kathir explained it as a case of tadmeen:
ولهذا ضمن يشرب معنى يروى حتى عداه بالباء
The word Yashrabu (to drink) includes (ضمن) the meaning of Yarwa (to quench one's thirst)
The translator of Ibn Kathir didn't translate the final part حتى عداه بالباء, "so that it could be used with with".
So the Quran said يشرب ("drink"), but uses the incorrect preposition and that is explained as being caused by the word يروى being invisibly inserted into the word يشرب and since يروى can be used with "with", then "drink with" is also correct.
"Sexual intercourse to your wives" (الرفث إلى نسائكم)
This is 2:187 translated by Muhsin Khan:
It is made lawful for you to have sexual relations (الرفث) with (إلى) your wives on the night of As-Saum (the fasts)..
The word إلى means "to". The word "with" would be مع or ب. It is not correct in Arabic to say الرفث إلى ("sexual intercourse to [someone]") as is found in the Qur'an. ِThe explanation give is that this is a case of tadmeen (insertion)  and that the word الإفضاء (to go) was inserted into the word الرفث ("sexual intercourse") and since it's correct to say الإفضاء إلى ("go to"), then الرفث إلى ("sexual intercourse to") is also correct.
The translation Sahih International translates it as "go to" and puts "for sexual relations" in brackets, although in Arabic the verb "go" is not present at all and the word "sexual intercourse" is:
It has been made permissible for you the night preceding fasting to go to your wives [for sexual relations]..
Tadmeen of prepositions themselves
Some scholars also say that, in addition to verbs, tadmeen can also be done with prepositions themselves.
[It is] the Day they will be tormented over (على) the Fire
The preposition "over" is said to do tadmeen of the preposition "in" (في), so it's "in the fire".
O you who have believed, when [the adhan] is called for the prayer on (من) the day of Jumu'ah [Friday], then proceed to the remembrance of Allah and leave trade. That is better for you, if you only knew.
The word من means "from". The explanation is that the correct preposition في was invisibly inserted into it.