y such that L(y) and y = x.
|
|   x in L iff there exists y in L such that y = x.

Since x and y are k-tuples, we bring in
the definition of equality for k-tuples:

|   x = y
|
|   iff
|
|   Proj&lt;j&gt;(x) = Proj&lt;j&gt;(y) for j = 1 to k.

Just to be tricky, hopefully without a self-deception --
but that all depends on what it is we wish to believe --
we exploit a standardized abuse of notation and write:

|   L(x&lt;1&gt;, ..., x&lt;k&gt;)
|
|   iff
|
|   There exists y such L(y) and Proj&lt;j&gt;(y) =