Straight Line Segment is Shortest Path between Two Points/Lemma 1

Lemma to Straight Line Segment is Shortest Path between Two Points
Let $p, q \in \R$ be real numbers such that $\sqrt {1 + p^2} = \sqrt {1 + q^2}$.

Then:
 * $1 + \size p = 1 + \size q$

Proof
By definition of absolute value of a real number:
 * $\sqrt {p^2} = \size p$

Hence: