Rearranging Sums, Differences and Two-Term Combinations in a Real Vector Space

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity and Real Normed, Inner Product and Hilbert Spaces: Notation, let $V$, with its addition and scalar multiplication, be a real vector space.

(i) Exchanging the middle terms. If $a,b,c,d\in V$, then

$$(a+b)+(c+d)=(a+c)+(b+d).$$

(ii) Translation invariance of differences. If $x,y,b\in V$, then $(x+b)-(y+b)=x-y$.

(iii) Difference of two sums. If $x,y,a,b\in V$, then

$$(x+y)-(a+b)=(x-a)+(y-b).$$

(iv) Cancelling an added vector. If $a,w\in V$, then $(a+w)-a=w$.

(v) Combinations of sums. If $t,s\in\mathbb{R}$ and $p,q,p',q'\in V$, then

$$t(p+q)+s(p'+q')=(tp+sp')+(tq+sq').$$

(vi) Combinations with weights summing to one. If $t,s\in\mathbb{R}$ with $t+s=1$ and $x\in V$, then $tx+sx=x$.