vector spaces: the proof that the sum of two subspaces is another subspace

$ U_1, U_2 $$ ⊂V $ They will be subspaces of V (a vector space). Define the sum of the subspace of $ U_1, $ Y $ U_2 $ It is defined as the set:

$ U_1 + U_2 $ $ = $ {$ u_1 + u_2: u_1 ∈ U_1, u_2 ∈ U_2 $}

Leave $ A $ denote the whole $ U_1 + U_2 $

A is a subspace if it meets all the criteria of a subspace, that is, $ 0∈A $, remains closed by addition, and remains closed by multiplication.

As $ U_1 $ It is a subspace, by definition of subspaces it contains. $ au_1 $ ($ a∈R $) $ 0 $ when a is equal to zero, and $ u_1 + w_1 $ (when $ w_1∈U_1 $).

As $ U_2 $ It is a subspace, by definition of subspaces it contains. $ au_2 $ ($ a∈R $) $ 0 $ when a is equal to zero, and $ u_2 + w_2 $ (when $ w_2∈U_2 $).

$ 0u_1 + 0u_2 = 0 (u_1 + u_2) = 0 $; it is an element of $ A $

$ au_1 + au_2 = a (u_1 + u_2) $; it is an element of $ A $

$ (u_1 + u_2) + (w_1 + w_2) = (u_1 + w_1) + (u_2 + w_2) $ it is an element of $ A $

Q.E.D.

This is my test, is it correct in a logical, symbolic, etc., has no clarity, format / structure, etc.?

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