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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