How to proof the distributive law in vectors?

Updated: 12/18/2022
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If we're talking vectors in a vector space, then distributivity is taken as an axiom. We can prove it for vectors in Rn, therefore (partially) proving that our common notion of vectors actually satisfies the axioms of a vector space. Below, a and b are scalars and v and w are vectors. a(v+w) = a( (v1, ..., vn) + (w1, ..., wn) ) = a(v1+w1, ..., vnwn) = (a(v1+w1), ..., a(vn+wn)) = (av1+aw1, ... , avn+awn) = (av1, ..., avn) + (aw1, ..., awn) = av+aw (a+b)v = (a+b)(v1, ..., vn) = ((a+b)v1, ..., (a+b)vn) = (av1+bv1, ..., avn+bvn) = (av1, ..., avn) + (bv1, ..., bvn) = av+bv

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Q: How to proof the distributive law in vectors?
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