Is writing "However it is a general fact that..." a valid statement in a proof? Martin Leibeck in his textbook A concise introduction to pure mathematics (third edition) brings an example of proof by contradiction in the beginning of the textbook (p.g. 6): (emphasis mine)

Example 1.4
No real number has a square equal to $-1$.
Proof:
Suppose the statement is false. This means there is a real number, say $x$, such that $x^2=-1$. However it is a general fact  about real numbers that a square of any real number is greater than or equal to $0$. Hence $x^2 \ge 0,$ which implies that $-1 \ge 0$. This is a contradiction.

My question:
Does the Italic section of the proof have valid steps necessary for proving such a statement? Is saying, "However it is a general fact about..." a valid step in writing a proof?  Wouldn't it be necessary to define what a "real number" is?
 A: Doing so would be extremely tedious and greatly slow down the process of learning new things. Most math books tell you who the intended reader is, this is so that you have the appropriate prerequisites to be able to fill gaps in proofs.
A: "General" seems to mean "well-known," and yes, with that meaning the statement is valid in a proof provided the fact is indeed well-known. When learning, you can so describe stuff that appears in textbooks for prerequisite courses.  When it's the kind of stuff that appears in the textbook for the course you're currently in, it's better to give a cite to a place in the textbook or a theorem name (e.g., Lebesgue Dominated Convergence).
A: No.  And in this case it's especially egregious as this prove seems to boil down to:  proof that $x^2\ne -1:$  c'mon, you know it is positive!
BUT note this is an example.  Not actually a theorem.  The purpose of this is not to show a rigorous proof, but simply to introduce the student to the concept of a proof by contradiction.  Rather than proving something significant, they showed an example where the student really already knew why.  This is so it is the concept of contradiction that is being shown.  Not the rigor.
I often use silly examples.  Such as Proof that I do not own a unicorn:  assume I own a unicorn.  Everyone who owns a unicorn is extremely happy. Therefore I am happy. But look at me; I'm obviously miserable.  So it'd be impossible for me to own a unicorn.
That's not really a valid argument.  All the statements are presumed as given and not verified.  
But it shows the idea that we presume the opposite, we deduce, we come to a contradiction.  So the opposite is impossible.
That's all that is relevent.
A: No, it is not. Nothing has been proven in the proof you posted. The general fact should be replaced with something like:
Every square of a real number is positive, because


*

*if $x >0$ then $x \cdot x > 0$

*if $x <0$ then $x \cdot x = (-y)(-y) >0$ for $ x \space \epsilon \space \mathbb{R}$ and $y = -x$

