Although Russia claims it has won control of Ukraine's eastern city of Bakhmut, after a grinding nine-month conflict in which tens of thousands of fighters have died, top Ukrainian military leaders say the battle is not over.

Ukrainian officials acknowledge they now control only a small part of Bakhmut.

But, Ukraine says, their fighters' presence has played a key role in their strategy of exhausting the Russian military. And they say their current positions in the areas surrounding Bakhmut will let them strike back inside the 400-year-old city.

"Despite the fact that we now control a small part of Bakhmut, the importance of its defense does not lose its relevance," said Col.-Gen. Oleksandr Syrskyi, the commander of ground forces for the Ukrainian Armed Forces. "This gives us the opportunity to enter the city in case of a change in the situation. And it will definitely happen."

The fog of war made it impossible to confirm the situation on the ground in Bakhmut. Russia's defense ministry said Wagner mercenaries backed by Russian troops had seized the city, but Ukraine President Volodymyr Zelenskyy said Bakhmut was not being fully occupied.

In a video posted on Telegram, Wagner head Yevgeny Prigozhin claimed the city came under complete Russian control at about midday Saturday. Holding a Russian flag before a group of at least nine masked fighters in body army who were toting heavy weapons, Prigozhin proclaimed: "This afternoon at 12:00, Bakhmut was completely taken."

More important for Ukraine has been the high numbers of Russian casualties and sapping of the morale of enemy troops for the the small patch of the 1,500-kilometer (932-mile) front line as Ukraine gears up for a major counteroffensive in the 15-month-old war.

"The enemy failed to surround Bakhmut. They lost part of the heights around the city. The continuing advance of our troops in the suburbs greatly complicates the enemy's presence," said Hanna Maliar, Ukraine's deputy defense minister. "Our troops have taken the city in a semi-encirclement, which gives us the opportunity to destroy the enemy."

About 55 kilometers (34 miles) north of the Russian-held regional capital of Donetsk, Bakhmut was an important industrial center, surrounded by salt and gypsum mines and home to about 80,000 people before the war, in a country of more than 43 million.

The city, named Artyomovsk after a Bolshevik revolutionary when Ukraine was part of the Soviet Union, was known for its sparkling wine produced in underground caves. It was popular among tourists for its broad tree-lined avenues, lush parks and stately downtown with imposing late 19th century mansions. All are now reduced to a smoldering wasteland.

Fought over so fiercely by Russia and Ukraine in recent months has been Bakhmut's urban center, where Ukrainian commanders have conceded Moscow controlled more than 90%. But even now, Ukrainian forces are making significant advances near strategic roads through the countryside just outside, chipping away at Russia's northern and southern flanks by the meter (yard) with the aim of encircling Wagner fighters inside the city.

Ukrainian military leaders say their months-long resistance has been worthwhile because it limited Russia's capabilities elsewhere and enabled Ukrainian advances.

"The main idea is to exhaust them, then to attack," Ukrainian Col. Yevhen Mezhevikin, commander of a specialized group fighting in Bakhmut, said Thursday.

Russia has deployed reinforcements to Bakhmut to replenish the lost northern and southern flanks and prevent more Ukrainian breakthroughs, according to Ukrainian officials and outside observers. Russian President Vladimir Putin badly needs to claim victory in Bakhmut, where Russian forces have focused their efforts, analysts say, especially after a winter offensive by his forces failed to capture other cities and towns along the front.

Ukraine's tactical gains in the rural area outside urban Bakhmut could be more significant than they seem, according to some analysts.

"It was almost like the Ukrainians just took advantage of the fact that, actually, the Russian lines were weak," said Phillips O'Brien, a professor of strategic studies at the University of St. Andrews. "The Russian army has suffered such high losses and is so worn out around Bakhmut that ... it cannot go forward anymore."

Ukrainian forces in the outskirts of Bakhmut and in the city bore relentless artillery attacks until a month ago. Then, Ukrainian forces positioned south of the city spotted their chance for a breakthrough after reconnaissance drones showed the southern Russian flank had gone on the defensive, Col. Mezhevikin said.

After fierce fighting for weeks, Ukrainian units made their first advance in the vicinity of Bakhmut since it was invaded nine months ago.

In all, nearly 20 square kilometers (8 square miles) of territory were recaptured, Maliar said in an interview last week. Hundreds of meters more have been regained almost every day since, according to Serhii Cherevatyi, spokesman for Ukraine's Operational Command East.

"Previously we were only holding the lines and didn't let Russians advance further into our territory. What has happened now is our first advance (since the battle started)," Maliar said.

Victory in Bakhmut does not necessarily bring Russia any closer to capturing the Donetsk region - Putin's stated aim of the war. Rather, it opens the door to more grinding battles in the direction of Sloviansk or Kostiantynivka, 20 kilometers (12 miles) away, said Kateryna Stepanenko, a Russia analyst at the Institute for the Study of War, a U.S.-based think tank.

Satellite imagery released this week shows infrastructure, apartment blocks and iconic buildings reduced to rubble.

In the last week, days before Russia announced the city had fallen into their control, Ukrainian forces retained only a handful of buildings amid constant Russian bombardment. Outnumbered and outgunned, they described nightmarish days.

Russia's artillery dominance was so overwhelming, accompanied by continuous human waves of mercenaries, that defensive positions could not be held for long.

"The importance of our mission of staying in Bakhmut lies in distracting a significant enemy force," said Taras Deiak, a commander of a special unit of a volunteer battalion. "We are paying a high price for this."

The northern and southern flanks regained by Ukraine are located near two highways that lead to Chasiv Yar, a town 10 kilometers (6 miles) from Bakhmut, that serve as key logistics supply routes. One is dubbed the "road of life."

Ukrainian forces passing this road often came under fire from Russians positioned along nearby strategic heights. Armored vehicles and pickup trucks driving toward the city to replenish Ukrainian troops were frequently destroyed.

With those high plains now under Ukrainian control, its forces have more breathing room.

"This will help us design new logistic chains to deliver ammunition in and evacuate the injured or killed boys," said Deiak, speaking from inside Bakhmut on Thursday, two days before Russia claimed control of the city. "Now it is easier to deliver supplies, rotate troops, (carry out) evacuations."

From The Associated Press