"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?.

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.
