"Exactly Two" or "Exactly k" from English into (Quantificational) Logic The inspiration is Example 2.2.3 #2(d) on P71 of How to Prove It by Daniel Velleman.

Analyze the logical forms of the following statement:
The number $x$ has exactly $k$ $n$th roots.
Answer : $\color{#FF4F00}{\exists \, r_1 \cdots \exists \; r_k} {\huge{[}}\, \color{#007FFF}{r_1 \;\& \, \cdots \& \,r_k \text{ are $n$th roots of $x$}} \; \text{ and } \; \color{green}{ r_1 \neq \cdots \neq r_k} \quad \text{ and }\color{#960018}{\text{ nothing else is a $n$th root of $x \;$}}{\huge{]}}  $
$= \color{#FF4F00}{\exists \, r_1 \cdots \exists \; r_k} {\huge{[}}\, \color{#007FFF}{r_1^n = x \;\& \, \cdots \& \, r_k^n = x \text{ are $n$th roots of $x$}} \; \wedge \; \color{green}{r_1 \neq \cdots \neq r_k}  \quad \wedge \;\color{#960018}{ \lnot \, \exists y \, {\Large{[}} \,y^n = x \, \wedge \, y \neq r_1 \, \wedge \cdots \wedge \, y \neq r_k \, {\Large{]}}} \;{\huge{]}} $.

I understand the necessity of the blue and green statements.
However, why is the (carmine) red necessary? At the beginning of each sentence, in orange, I declared the existence of only $k$ variables (ie $\color{#FF4F00}{r_1, ..., r_k}$), so there are simply no more variables that could serve as the $(k + 1), (k + 2), ...$ variables. Thus, how and why does the orange NOT imply the red tacitly and wordlessly?
I referenced How to convert an English sentence that contains "Exactly two" or "Atleast two" into predicate calculus sentence?.
 A: Your orange, blue and green parts declare that there are at least $k$ n'th roots. However it is perfectly possible that there are more than $k$ \emph{objects} which the variables are currently standing in for. That is what your red sentence excludes.
For instance $\exists x (x^2=1)$ says that $1$ as a square root (e.g. 1) however $-1$ could also stand in here.
A: Let's take a simpler case. Suppose you want to render 'There is exactly one F' into standard logical notation.
The standard answer would be

$\exists x(Fx \land \forall y(Fy \to y = x))$

or some equivalent like

$\exists x(Fx \land \neg\exists y(Fy \land y \neq x))$.

Now imagine someone asking what the clause after the conjunction was doing. "Why is it necessary? At the beginning of the sentence I declared the existence of only one variables (i.e. $x$), so  and why doesn't  the first conjunct imply the second one tacitly and wordlessly?". Well it should be obvious in this case what's gone wrong. If we write down only the first conjunct, i.e. just write

$\exists xFx$

this tells us that at least one thing (in the relevant domain) is F. That's what the existential quantifier means. It doesn't declare that there is only one $F$ (though of course it only uses one variable). That is to say, it doesn't rule out there being more than one F; we need the extra clause precisely to rule that out. 
That should give a clue, then, why you need the "red" clause in the more complex case.
A: The importance of the red part may be more clear with the equivalent phrasing

If $y$ is an $n$-th root of $x$, then $y=r_1$ or $y=r_2$ or .. or $y = r_k$.

A: as per your comment (above): 
The orange, blue, and greed section states the existence of $k$ distinct roots.  The red statement says that there are no others.  Thus there are EXACTLY $k$ roots.  The red statement formalizes the meaning of the work exactly in the statement "The number $x$ has exactly $k$, $ n$-th roots.
