YONGSAN GARRISON, South Korea — Vice President Dick Cheney — greeted in Collier Field House on Friday by servicemembers chanting “USA, USA!” — trumpeted U.S. efforts in Iraq while complimenting South Korea’s troop commitment.

Cheney, at the end of a weeklong trip that included visits to Japan and China, praised actions in Iraq and Afghanistan, saying the Bush administration “will take the fight to the enemy and we will win.”

“We will destroy the remnant of violence of oppressive regimes, and together, we will win this essential victory in the war on terror,” Cheney said.

In earlier comments to reporters, Cheney acknowledged “the price of leadership is high” but “we know our cause is just.” He did not take questions.

South Korea plans to send about 3,600 troops to Iraq, but violence there has flared domestic opposition. It sent medical and engineer soldiers to Afghanistan, and the Iraq dispatch would place the country just behind the United States and Britain in the number of deployed troops.

Action by the United States and other allies in Iraq put “a state sponsor of terror” out of business, Cheney said.

“A year ago, Saddam Hussein controlled the lives and the future of almost 25 million people. He has pursued weapons of mass destruction and threatened more. Tonight, he is in jail,” he said to cheers.

In regard to North Korea, Cheney said the administration supports six-nation talks to stop its nuclear program. The United States stands with South Korea for a “complete, verifiable, irreversible dismantlement of North Korea’s nuclear programs.”

Cheney said he met Friday with acting President Goh Gun, who’s leading the country after President Roh Moo-hyun’s impeachment.

While the administration would like to see reconciliation between the Koreas, “the stability of this peninsula is built on a brave alliance between the Republic of Korea and the United States,” Cheney said.

The visit was Cheney’s first to South Korea since he was secretary of defense under former President George H.W. Bush. He reminded soldiers that the current President Bush has supported three military pay raises and continues to ensure troops have the best equipment.

After finishing his remarks in Collier Field House, which was decked out like a bunker with camouflage netting, Cheney re-enlisted 11 servicemembers, handing each a ceremonial coin.

“My unit asked me if I would like to re-enlist today, and I said ‘yes,’” said Sgt. Miranda Jackson, who works in aviation operations at Yongsan Garrison. “I wouldn’t miss this opportunity to do it. It’s awesome. It’s a one-time opportunity.”

Cheney’s wife, Lynne, gave a 30-minute presentation to children at Seoul American Elementary School, reading from her book, “America, A Patriotic Primer.” She spoke about the founding fathers and American ideals.

Her presentation also was heard by students in Daegu and Pusan, who were linked to her speech through video teleconferencing equipment. Her presentation focused on the importance of learning history and American concepts.

“We are so fortunate to be Americans, and the whole world is fortunate that America exists because America has spread freedom to so many people,” she said.

She took questions from several children. One student asked if her husband fought in a war.

“No, he didn’t,” she said, shortly before pointing out a U.S. soldier who fought in Vietnam in her history book. “When he was a young man, we lived in the 1960s. He (Dick Cheney) was in college, and so he did not fight in the war.

“But he has served as secretary of defense, and he and I are both proud — so proud — that we can be associated with the people who are protecting and defending our country now. We appreciate the service of your parents.”