Two reduced expressions of one element whose subword descriptions agree

For (i): $v=2431=s_1{s}_2{s}_3{s}_2=s_1{s}_3{s}_2{s}_3$, because the two words differ only in their last three letters, which are the two words $s_2{s}_3{s}_2$ and $s_3{s}_2{s}_3$ of the same element of the rank-two parabolic $⟨s_2,s_3⟩$ (the braid relation), and $v$ has inversion number $4$ by [F3], so both words have length $4=\mathrm{\ell }(v)$ and are reduced expressions of $v$ by [F1]; the product $s_1{s}_2{s}_3{s}_2=2431$ is computed by applying [F2] letter by letter. The element $u=s_2$ has one-line form $1324$ and $\mathrm{\ell }(u)=1$; it occurs in $s_1{s}_2{s}_3{s}_2$ at positions $2$ and $4$ and in $s_1{s}_3{s}_2{s}_3$ at position $3$ only, so the subword criterion [F4] gives $u\le v$ from either occurrence, at different positions in the two expressions of $v$.

For (ii): fix either reduced expression of $v$. Every subword value $x$ is the product of a subword of that reduced word, hence $x\le v$ by the right-to-left direction of [F4], and $1\le x$ by [F6], so the set of subword values of either expression is contained in the interval $[1,v]$; conversely, by [F5] every $x\in [1,v]$ has a reduced subword expression inside every reduced expression of $v$, hence is a value of a subword of either of the two words. Therefore the sets of subword values of the two expressions are both equal to $[1,v]$, so they coincide. Enumerating the $16$ subwords of $s_1{s}_2{s}_3{s}_2$ by multiplying out the indicated letters with [F2] gives exactly the $12$ displayed permutations $1234$, $1243$, $1324$, $1342$, $1423$, $1432$, $2134$, $2143$, $2314$, $2341$, $2413$, $2431$ (the empty subword gives $1234$), and enumerating the $16$ subwords of $s_1{s}_3{s}_2{s}_3$ gives the same $12$ values; hence $[1,v]$ is exactly the displayed set.

Collecting: the two reduced expressions of $v$ describe the same interval below $v$, both by the general equivalence of [F5] and by the explicit enumeration of step 1.2 of the $16$ subwords of each expression; the element $u=s_2$ is described at position $2$ in the first expression and position $3$ in the second, so the positions may differ while the value is the same, as (i) says. If the two descriptions could disagree, then some element would have a subword expression in one reduced expression of $v$ and none in the other, contradicting [F5]; all computations are finite and use no choice principle.