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A troubled city and school system now have even more trouble.

The fight is on, putting 300,000 Chicago public school children out of their classrooms.

The dispute between Chicago school managers and the city’s teachers union has been brewing for weeks before Thursday, when it became official.

How long it will last is impossible to say. It’s certain there will be no settlement anytime soon unless one side or the other does in private what it promised in public that it will never do.

Union president Jesse Sharkey said teachers are demanding “dignity” and “respect.” That means they want a lot more money that Chicago Mayor Lori Lightfoot insists the city doesn’t have.

In addition to the 25,000 striking teachers, 7,000 support workers who belong to the Service Employees International Union also have walked off the job. They, too, are in search of respect that comes in the form of cold, hard cash.

But this dispute is not entirely about pay and benefits, it’s also about politics.

The teachers union is trying to do to Mayor Lightfoot what it did to her predecessor, Rahm Emanuel.

It crushed Emanuel with its 2012 walkout. Now the union intends to show Lightfoot who’s boss, something that’s particularly important after Lightfoot earlier this year soundly defeated the union’s choice for mayor, Cook County Board President Toni Preckwinkle.

Lost in this political equation are the K-12 students who are trapped in a school district widely recognized as subpar. A strike is something they and their parents, many of them low-income, don’t need.

The union has flatly rejected a contract offer that many consider generous, even overly generous. Lightfoot proposed a five-year contract with a 16 percent pay hike on top of step-increases based on longevity.

The union wants 15 percent over three years plus commitments to hire large numbers of nurses and school psychologists.

There’s a lot more to it, of course. But readers get the idea. Big money is on the line, an additional $2.5 billion in spending, according to Lightfoot’s estimate.

There’s always a lot of huffing and puffing when workers go out on strike. Union leaders demand more and promise to stay out until they get it while management insists that the contract offer the union rejected is fair, reasonable and affordable for taxpayers. Rhetoric aside, reaching a settlement always revolves around who has the most leverage.

The union has thrown the school year into chaos, and that’s a lot of leverage.

At the same time, Lightfoot said any school days lost will not be made up. If true, striking teachers could lose more than they would gain if and when management sweetened its offer. So management has some leverage, too.

As for the school system itself, it’s struggling. Chicago’s public schools have lost 75,000 students (15 percent of its student population) since 2000 and, like the city, is buried in debt and deficits.

It’s hard to imagine that it will come out of this strike in better shape than it was before the strike.Nonetheless, the die is cast.