Could someone help me by explaining how this is formed?

The question here refers to an answer to my question (second comment on
answer), but I don't understand it: [Could someone help with the gramatical
construction of this
phrase?](https://japanese.stackexchange.com/questions/42780/could-someone-
help-with-the-gramatical-construction-of-this-phrase)

Answer (3-㋒): <https://dictionary.goo.ne.jp/jn/120514/meaning/m0u/>

It looks like someone transformed a masu-form verb to a "masu-te"-form, which
I never heared of:

> する、します、しまして  
>  たべる、たべます、たべまして

If I am not mistaken, the referenced answer also talks about the potential
form. How does this relate? Furthermore, is this construction in any way
related to はじめまして?
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2019-09-05T14:28:47.937
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question
grammar て-form potential-form form
Can someone explain to me how "としまして" is formed?