In determining whether any segments intersect, why there must be some sweep where segments $a$ and $b$ are consecutive?

In CLRS, Section 33.1, we are given the any-segment-intersect algorithm. It’s a cool algorithm for sure but going through the correctness proof, I don’t know how to prove the following:

“Given that we have two segments (call them $ a$ and $ b$ ) that intersect at some point $ p$ , then there must be some sweep line $ x$ for which intersecting segments $ a$ and $ b$ are consecutive in the total preorder.” why they must consecutive at some sweep line?

I do see why visually using the figure on page 33.4 (page 1023) but I don’t know a proof for this statement. How can I prove this?

If anyone had read that section, please enlighten me. Thank you

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