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