Vector space example

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

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

Assert for any $x,y,x',y' \in \mathbb{R}$ ,

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

$x+x' = x'+x$ ,

$y+y' = y'+y$ ,

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

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

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

Assert for any $x,y,x',y',x'',y'' \in \mathbb{R}$ ,

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

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

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

$(x+x')+x'' = x+(x'+x'')$ ,

$(y+y')+y'' = y+(y'+y'')$ ,

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

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

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

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

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