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MANCHESTER, N.H. — Rudy Giuliani cheated on his wife with a married New Hampshire hospital administrator, multiple sources told The Post.

The mayor-turned-presidential lawyer, 74, and Maria Rosa Ryan, 53, began their affair before he and wife Judith Nathan separated last month — and Nathan filed for divorce five days after the pair was spotted getting cozy at a ­resort hotel in the Granite State.

Giuliani, who has recently been the face of President Trump’s legal team, denied the affair to The Post Tuesday after it broke the story of his relationship with Ryan.

He acknowledged that he and Ryan had dinner and watched a movie at the posh spa — but claimed he was already “in effect separated’’ from Nathan at the time.

But Nathan, 63, angrily shot back in a statement, “My husband’s denial of the affair with the married Mrs. Ryan is as false as his claim that we were separated when he took up with her.”

Nathan filed for divorce April 4 — after Giuliani publicly stepped out with his mistress on a March 29 tour of the hospital she runs. The visit was covered by local TV news cameras.

The former US attorney — who as head of global consulting and security firms has done business for nations including Qatar and Colombia, as well as OxyContin drug giant Purdue Pharma — told reporters he was touring the tiny, rural hospital to learn about its cybersecurity program.

Local ABC-TV affiliate WMUR aired video of ”America’s Mayor” walking with Ryan through her 25-bed Cottage Hospital.

“Giuliani said he hopes to take what he can learn in Woodsville and incorporate it into his cybersecurity firm,” WMUR reported.

Ryan crowed to the Caledonian-Record newspaper that Cottage “was among the first hospitals in the nation to adopt the newest technologies’’ — and “that’s why we get a lot of attention from people like Rudy Giuliani.”

But after the hospital tour, Giuliani and Ryan — who lives with her husband, Robert, a former Marine, in Manchester — headed an hour north to the Mountain View Grand Resort and Spa in Whitefield.

The thrice-married Giuliani booked the posh resort’s spacious Summit Suite — with views of the Presidential Range — for the pair, sources said.

Past guests at the historic, four-star spot include Presidents Theodore Roosevelt and Richard Nixon, author Stephen King and former first lady Michelle Obama.

Giuliani and Ryan dined with three male guests in the resort’s private dining area, The Hunt Room, on March 29, according to sources. He ordered the $32 beef rib.

After dinner, “the woman went back to his room with him,” a waitress told The Post.

“We were all surprised because he is really getting on in years, and she was quite a bit younger than him. We were all like, ‘Hmmm,’ ” the server added.

Hotel staff noted that while Giuliani didn’t seem as if he was trying to hide his girlfriend from them, he was careful to meet with Ryan only in more secluded areas of the resort.

A male assistant for the former New York City mayor made sure that his boss was on the hotel’s “VIP list’’ and arranged for him to watch “The Godfather” and “Godfather II” in its private movie-screening room.

“[Giuliani] was VIP, so when he requested the movies, I made sure to have them ready for him,” a hotel employee said. “We all recognized him right away. We were all very impressed.”

Staff described Giuliani as very friendly.

He even spent 45 minutes chatting with a bellman on the veranda before escorting Ryan back to his suite — with its king-size, wood-frame bed, spacious balcony and curated library with ­titles including “The American Patriot’s Almanac” and John ­Grisham’s “The Firm.”

Giuliani acknowledged to The Post, “I do know Dr. Ryan. She is a friend of mine, so is her husband, Bob. She’s a Ph.D. and a very, very fine woman.”

But when pressed about whether he had ever a sexual relationship with Ryan, he said, “I never spent the night with her. There’s no proof. There can’t be because we never did anything.”

Giuliani — who rose to national prominence thanks to his tough anticrime stance and highly visible role as the mayor who guided the Big Apple through 9/11 — joined Trump’s personal legal team this spring. Giuliani said he is focusing on special counsel Robert Mueller’s investigation into any possible ties between Trump’s 2016 campaign and the Kremlin.

But days after joining Team Trump, Giuliani jetted to Fort Lauderdale, Fla. — although hardly on government business.

He helped defend his mistress’ daughter, Vanessa Ryan, in court against charges of car-insurance fraud.

Giuliani told the Daily Mail that 32-year-old Vanessa, who was convicted of careless driving in 2016, according to court records, was his personal assistant.

Vanessa Ryan’s LinkedIn profile says she works for JetBlue. She is not listed on the employee roster for his security firm.

When asked if she worked for Giuliani and if she knew about the affair, Vanessa Ryan told The Post Tuesday, “Whatever story you’re trying to run is completely false, and you can quote me on that. Find a story elsewhere.”

Giuliani married Nathan in 2003, after his second wife, Donna Hanover, accused him of cheating and divorced him in 2002. He and Hanover had been married 18 years. His first wife was a second cousin, to whom he was married for 14 years.

When Giuliani confirmed his split from Nathan to The Post last month, he said, “In these divorce situations you cannot place blame, it is 50/50. There are problems on both sides.”

Meanwhile, he recently trashed Stormy Daniels — who has accused Trump of having an affair with her — as having “no reputation” because she’s a porn actress.

Ryan — who has a doctorate in health care administration from Warren National University, an online school that the feds have described as a “diploma mill’’ — was walking the family’s pit bull mix Tuesday when a Post reporter approached outside her home and asked about the affair.

The mom of three grown children called to her husband from the street. He told the reporter to “f–k off.”

When pressed for comment, his wife added, “Go ahead and do it, do what you normally do and ruin people’s lives.”

A day earlier, her husband had waved goodbye to his wife as she left for work, and when she returned home later, the pair snuggled on a chair on their porch.

Giuliani and Nathan don’t have a prenup and are fighting over an estimated $30 million fortune.

Additional reporting by Laurie Mizrahi and Kate Sheehy