LINCOLN, Neb. — Seven states trying to block part of the federal health care law that requires contraception coverage will continue with their lawsuit despite last month's U.S. Supreme Court ruling that upheld most of the law, according to Nebraska Attorney General Jon Bruning, who is leading the case.

The federal lawsuit is challenging a rule that requires contraception coverage in health care plans – including for employees of church-affiliated hospitals, schools and outreach programs. The suit argues that the rule violates the rights of employers that object to the use of contraceptives, sterilization and abortion-inducing drugs.

The U.S. Department of Justice wants the suit dismissed, in part because the president is trying to work out a compromise, but Bruning and his fellow Republican attorneys general in Florida, Michigan, Ohio, Oklahoma, South Carolina and Texas aren't backing down.

"This rule is a brazen violation of the First Amendment rights of millions of Americans," Bruning spokeswoman Shannon Kingery said. "We will continue to fight this attack on religious liberty."

Some legal experts said that even though the nation's highest court largely upheld the law, the lawsuit is narrowing in on a separate issue and has a decent chance.

Adam Samaha, a constitutional law professor at New York University's School of Law, said other recent Supreme Court rulings suggest the court has "some sympathies with religious organizations being burdened by government." He cited a unanimous decision earlier this year in which justices sided with a religious school in an employment discrimination lawsuit.

But he also noted that President Barack Obama's administration, in response to the criticism from religious groups, delayed enforcement of the provision until next summer and has said it would shift the requirement from employers to health insurers. Samaha said that shift bolsters the administration's position in the legal challenge.

"Everybody agrees that this is far from a frivolous suit," added Samaha's colleague, NYU law professor Richard Epstein. "Intellectually, it's a very powerful suit."

The lawsuit was filed by Bruning, who was running for U.S. Senate at the time, in U.S. District Court in Nebraska. Plaintiffs also include three Nebraska-based groups – Catholic Social Services, Pius X Catholic High School and the Catholic Mutual Relief Society of America – along with a nun and a female missionary.

Justice Department lawyers say the plaintiffs have failed to show they face the immediate threat of having to offer the coverage, because the federal government delayed enforcement until August 2013 so that the groups' concerns could be addressed. The agency said the attorneys general lack the legal grounds to sue over the provision because state governments don't enjoy the same First Amendment protections as individuals.

But the lawsuit argues that the rule will effectively force religious employers and organizations to drop health insurance coverage, which will raise enrollment in state Medicaid programs and increase patient numbers at state-subsidized hospitals and medical centers. The U.S. Department of Health and Human Services is named as a defendant.

Donald Blankenau, an attorney for an Omaha Catholic missionary who is a plaintiff in the lawsuit, said the Nebraska case focuses on different issues than those addressed in the Supreme Court's ruling last month.

"Had the court knocked out the mandate in the case, it probably would have resolved the issues in ours. But since it didn't, our set of issues will proceed," Blankenau said.

Obama administration officials have said they don't want to abridge anyone's religious freedom, but want to give women access to important preventive care. Supporters of the rule, including the American Civil Liberties Union and women's advocacy groups, say the measure is about female health.

Officials have said the Obama administration's ruling was carefully considered, after reviewing more than 200,000 comments from interested parties and the public. The one-year extension, they said, responds to concerns raised by religious employers about making adjustments.

Administration officials stress that individual decisions about whether to use birth control, and what kind, remain in the hands of women and their doctors.

Still, Samaha noted, challenging the requirements is a win-win for Republican attorneys general, particularly those from conservative states who would benefit from being seen as standing up for religious freedom.

"If they prevail in the constitutional challenge, then they get a judicial order that might help change the state of affairs," the law professor said. "If they lose, they've indicated whose side they're on, and not just with cheap talk."

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Sean Murphy reported from Oklahoma City.

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