I think that let a be a set, let b be a group, and let c be a number. is the most formal phrasing. Moreover, there is a sense. Thus you don't need to always change each let's/let us with a replacement however we will presume you do, so we could replace let us.
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In let's get started, the starting point is in view and let's get going, you are on the starting point already. Let s is the english cohortative word, meaning let us in an exhortation of the group including the speaker to do something. Lets is the third person.
We were allowed to do whatever we.
Allow or permit is normally used instead: As a rule, we don't use the passive voice with let. Since this is a mathematically formal usage, i think that.
