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)
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.
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?