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Proof without words: One house is temporarily deleted. The lines connecting the remaining houses with the utilities divide the plane into three regions. Whichever region the deleted house is placed into, the similarly shaded utility is outside the region. By the Jordan curve theorem, a line connecting them must intersect one of the existing lines.
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A sewer pipe is normally at neutral air pressure compared to the surrounding atmosphere.When a column of waste water flows through a pipe, it compresses air ahead of it in the system, creating a positive pressure that must be released so it does not push back on the waste stream and downstream traps, slow drainage, and induce potential clogs.
Vertical pipes, usually made of precast concrete, called manholes, connect the mains to the surface. Depending upon site application and use, these vertical pipes can be cylindrical, eccentric, or concentric. The manholes are used for access to the sewer pipes for inspection and maintenance, and as a means to vent sewer gases.
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Service laterals, where present, can be reconnected from within the newly-formed larger-diameter pipe, by cutting replacement openings using robotically controlled cutting devices, then sealed using specially-designed CIPP materials referred to as 'top-hats'. The resins used are typically polyester for mainline lining and epoxy for lateral lines.
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