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Harold Washington and Jane Byrne, who both have waged four-year campaigns for the privilege of attaching the title ”mayor” to their names, are entering the 10 final days of a battle that will make Chicago political history.

Washington, who for nearly 3 1/2 years was locked in a Chicago City Council power struggle that hindered his leadership but solidified his constituency, is seeking to break the one-term jinx that has felled every Chicago mayor since Richard J. Daley died in 1976.

Byrne, one of those mayors voted out of office after only one term, is trying to make a comeback in a town that was once described by a political scholar as a place that ”don`t back no losers.”

And, more or less on the sidelines, are the other major mayoral candidates: Ald. Edward Vrdolyak (10th), Cook County Assessor Thomas Hynes and Republican Donald Haider, who have little to win or lose in the Feb. 24 primary election. They are waiting for the April 7 general election.

Byrne`s political operatives say they don`t plan major strategy shifts in the campaign`s final days.

”She`ll be taking it to the streets,” said Joe Pecor, Byrne`s press secretary. ”She`ll be working wall to wall, citywide, in every neighborhood of the city, being where the people are.”

A television advertising blitz will supplement Byrne`s appearances, Pecor said.

He concedes that although Byrne plans to go to nearly every neighborhood, she will spend most of her time in the lakefront and Hispanic neighborhoods that remain ”swing” areas in next week`s primary.

Byrne will run an ”uplifting” campaign concentrating on ”positive”

aspects of her administration and city life, Pecor said. But as late as Friday, her television commercials continued to show a Chicago skyline split asunder. A new spot will sock Washington for receiving campaign contributions from collection agencies that have been mentioned in investigations of city government corruption.

With an eye to the important ethnic vote, Byrne also last week taped a commercial appealing directly to the city`s Greek community, a traditionally generous contributors` bloc, to be played during Greek programming on WCIU-TV (Channel 26) in the next week.

The highlight of her campaign, Byrne said, will be a 30-minute ”debate” to be televised at 5:30 p.m. Sunday on WGN-TV (Channel 9). Byrne`s show reportedly will be a series of videotaped questions in a ”man-on-the-street” form, followed by Washington answers culled from taped appearances of the past and seemingly ”live” answers from Byrne. The taped show, completed Friday, cost $8,000 to produce and will cost $17,000 to air.

Byrne`s ”debate” follows six weeks devoted to a debate on debates, with Byrne and Washington each contending that the other set up impossible demands and secretly wanted to avoid a head-to-head confrontation.

Both were correct in their charges. Byrne had refused to have a studio audience, to avoid spectator response that had jarred her in debates four years ago and to avoid the generally accepted belief that Washington, in the live situation he demanded, would appear funnier and warmer than the often-dour image Byrne projects.

Washington`s campaign will make use of Byrne`s record as mayor, a strategy the mayor has used repeatedly during his campaign.

”Being in front makes you work harder because you`re on the offensive and defensive,” said Kenneth Glover, Washington`s campaign manager, reflecting on new polls that show the mayor ahead.

”If I could wish for this campaign, it would be the ability to focus on `Do you like what the mayor`s done?` and compare the two mayors` records,” he said.

When Washington began his television ad campaign several weeks ago, he stressed his achievements as mayor. But more recently, he has shifted his campaign to attacks on Byrne and her record.

In the final week, his chief strategists say, he will feature a combination of the two, with some ads stressing what he has done for Chicago and others stressing what Byrne did to the city.

And though Washington has spent more campaign time in supposedly

”unfriendly” areas, such as the Northwest and Southwest Sides, than Byrne has spent in black neighborhoods on the South and West Sides, during the last days he will turn more to his base of support in black neighborhoods.

For example, in what turned out to be a campaign issue exploited by Byrne last week, more than 100 volunteers from the South promise to arrive by caravan from Selma, Ala., under the sponsorship of the United Church of Christ`s Commission for Racial Justice.

The volunteers plan to work to get out the vote for Washington for the rest of the primary campaign.

Byrne has described the volunteers as part of the ”outside forces”

Washington has been using, including campaign funds he raised on a visit last week to Atlanta.

And next weekend, days before the primary, members of the Congressional Black and Hispanic Caucuses are expected to converge on Chicago to campaign for the mayor throughout the city, but primarily in black and Hispanic areas. The third Democratic primary candidate is Sheila Jones, the disciple of political extremist Lyndon LaRouche, who has waged a low-profile campaign, emerging from a fairly secretive political structure only occasionally to spread her political leader`s philosophies about international trade, drug problems and the proliferation of acquired immune deficiency syndrome.

Haider, who faces token challenges from three opponents in the Republican primary, said that in spite of his generally accepted shoo-in status, his workers will be calling or sending campaign literature to about 7,500 hard-core Republican households to let them know that he is in the race.

”We`re taking nothing for granted,” said Haider`s press secretary, Brian Sweeney. ”We know this is the land of the La Rouchies.”

Though Vrdolyak also faces only the challenge of a write-in candidate and is the only mayoral candidate on the Illinois Solidarity Party ballot, his aides say this week will be much like the earlier ones in the campaign:

pushing for Byrne`s nomination, attacking what he calls the ”Johnny-come-lately” entry by Hynes on the third-party Chicago First label and stressing that he would be the best candidate in the general election.

Hynes, the only major candidate who can`t be found the primary ballot, this week will begin what Robert Benjamin, his press secretary, calls his

”soft-shoe offstage.”

”We`re going to get out of the way almost entirely,” Benjamin said.

”During these 10 days, Hynes is not going to be presenting himself at all.”

The same cannot be said in the final 10 days for Washington and Byrne, because for one of them the ”City Hall `87” campaign ends Feb. 24.