maximal independent set on grid graph proof
As we define v′ if it is in T and is in S

I think you're misinterpreting the definition of $v$:

For each $v \in T$, define $v'=v$ if $v\in S$ or as its neighbor in $S$ otherwise

$v'$ is always defined; it's just a question of what it's defined to be. The above definition means that, if $v \in S$, $v'$ is defined to be $v$, and if $v \notin S$, $v'$ is defined to be $v$'s greater-or-equal-weight neighbour in $S$ (the existence of which was established in part 1a). Then $w(v) \le w(v')$ for each $v \in T$ by construction.
(This is the only way I see to parse the definition that makes grammatical sense, though I think it could be worded better, to prevent people reading &quot;define $v'=v$ if $v\in S$&quot; as a &quot;parenthesised unit&quot;.)