Vector space example

We present a few components of an algbraic proof that $\R^2$ is a vector space (using the fact that $\R$ is a vector space).

Assume for all $x,y,x',y' \in \R$ , $(x, y) + (x', y') = (x + x', y + y')$ .

Assert for any $x,y,x',y' \in \R$ ,

(x, y) + (x', y') = (x + x', y + y')

,

x + x' = x' + x

,

y + y' = y' + y

,

(x + x', y + y') = (x' + x, y' + y)

,

(x' + x, y' + y) = (x', y') + (x, y)

,

(x, y) + (x', y') = (x', y') + (x, y)

.

Assert for any $x,y,x',y',x'',y'' \in \R$ ,

(x, y) + (x', y') = (x + x', y + y')

,

(x + x', y + y') + (x'', y'') = ((x + x') + x'',(y + y') + y'')

,

((x, y) + (x', y')) + (x'', y'') = ((x + x') + x'',(y + y') + y'')

,

(x + x') + x'' = x + (x' + x'')

,

(y + y') + y'' = y + (y' + y'')

,

((x + x') + x'',(y + y') + y'') = (x + (x' + x''), y + (y' + y''))

,

(x, y) + :(x' + x'', y' + y'') = (x + (x' + x''), y + (y' + y''))

,

(x + (x' + x''), y + (y' + y'')) = (x, y) + (x' + x'', y' + y'')

,

(x' + x'', y' + y'') = (x', y') + (x'', y'')

,

((x, y) + (x', y')) + (x'', y'') = (x, y) + ((x', y') + (x'', y''))

.