Can this be done in a simplier way? Magic Squares

Main Question or Discussion Point

Proving Axiom 1 of all 3 x 3 magic squares.

I used summation notation to do so but it is extremely long and cumbersome.

I attached the pdf file with the work.

Is there a way to do this in a simpler more concise manner?

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What is this axiom 1 that you're trying to prove? That PDF is impenetrable.

To form the basis, consider the equations you have regarding some $$A$$ a magic square, denote $$x_i = a_{1i} \in A \quad 1 \le i \le 3$$ and $$x_i = a_{2i} \in A \quad 4 \le i \le 6$$ and so on, then:

$$x_1 + x_2 + x_3 = x_4 + x_5 + x_6$$
$$x_1 + x_2 + x_3 = x_7 + x_8 + x_9$$
$$x_1 + x_4 + x_7 = x_2 + x_5 + x_8$$
$$x_1 + x_4 + x_7 = x_3 + x_6 + x_9$$
$$x_1 + x_2 + x_3 = x_1 + x_4 + x_7$$

If you play with these equations for a bit you can get the basis matrices.