Egyptian President Hosni Mubarak, in what some politicians call a crucial reassertion of his authority in dealing with Islamic fundamentalists, has banned a demonstration here Friday and ordered police to deal "firmly" with anyone who marches in defiance of the ban.

But Sheik Hafez Salama, the 60-year-old fundamentalist religious leader who called the march in support of immediate application of Islamic law here, has insisted that the demonstration will be peaceful and that it will take place.

Although an overwhelmingly Moslem nation, Egypt has a western-style government and legal system, and religious fundamentalists have been pressing for implementation of much stricter Islamic law.

In an interview earlier this week, while consultations with the government were still going on, Salama said, "We will move ahead with our peaceful march. There is no problem between us and the government or the security police. We have invited all the people to join this march -- including the security police."

Members of the staff at Salama's Al Nur mosque declined today to say whether the ban will be defied. But this is the last Friday -- the Moslem day of rest -- in the holy month of Ramadan. The mosque staff said they expected large crowds of worshippers -- and police.

Mubarak has left no room for doubt about how his government views the potential challenge to its authority by Salama's group or others.

The lead story in all official and semiofficial newspapers yesterday morning was the Interior Ministry's warning against "continued attempts to whip up the feelings of the masses" and the ban on demonstrations.

Today, Mubarak was quoted in stories under bannered headlines as denouncing "erroneous democratic practices" and "one-upmanship" that could have serious effects in the "grave stage through which we are now passing." He called for "reason, calm, and a gradual approach" to resolving the heated debate over fundamentalist demands.

Even if Salama backs down, however, politicians and intellectuals here who support secular government or liberal interpretations of Islamic law are increasingly nervous about the renewed wave of fundamentalist activism.

In a series of interviews during the past two weeks they have talked not so much of the Islamic movement's strength as of the government's apparent weakness in dealing with it, especially on the crucial question of sharia, or Islamic law.

"If they feel you are weak, they will eat you," said Farag A. Fouda, an outspoken advocate of secular government who resigned from the New Wafd Party when it joined forces last year with the relatively moderate fundamentalist Moslem Brotherhood. "We are just on the edge of being pushed into making Egypt a religious state."

A few, such as political writer and Islamic scholar Hussein Ahmad Amin, warn that "everybody is expecting the extremists to take over -- the box office is open for seats in the new regime."

In the form advocated by its most absolutist proponents, application of Islamic law would deeply affect everything from banking to marriage to criminal justice. Under strict sharia, the payment of interest is forbidden, taxes are replaced by tithing, alcoholic beverages are banned, adulterers stoned to death and the hands and feet of thieves amputated.

None of these provisions is now in effect here, nor do many people expect them to be in the near future, but 90 percent of the population is Moslem, and across the political spectrum lip service, at least, is paid to the principles of sharia.

"The government cannot, as good Moslems, refuse the sharia," said sociologist Saadeddin Ibrahim, "and meanwhile they do not really want to apply it."

Mubarak, whose predecessor Anwar Sadat was assassinated by a violent Moslem fringe group in 1981, has attempted to open traditionally authoritarian Egyptian politics to democratic debate and dialogue.

Ibrahim suggests that some of the fears heard in intellectual circles are misplaced. "I'd rather see the groups acting in the open and acting legally and lobbying for issues peacefully," he said, "than having them as social dynamite hanging over our heads."

But the question of sharia is so sensitive that even though the government party has an 85 percent majority in the legislature, the People's Assembly, it has not wanted to bring the issue of sharia to a vote.

A motion to do so was offered on May 4, and the assembly's speaker, Rifaat Mahgoub, has talked vaguely since then about "cleansing" laws that might conflict with Islamic principles.

"It is not a big issue," Mahgoub said earlier this week. "We will not change any law without taking into consideration all the concerns and wishes of all our people."

One part of the society gravely concerned with the issue is the Coptic Christian minority, whose 6 million people long have felt vulnerable to fundamentalist Islamic fervor.

"The Copts will be automatically demoted to second-class citizens if sharia is enacted," one Moslem political analyst said.

A spokesman for Coptic Pope Shenouda III at his cathedral, located near the Al Nur Mosque, said today that in the current environment "his holiness tries to avoid speaking on such critical matters."

"His excellency was in exile for 40 months," the spokesman noted. Shenouda was accused then by the government of exacerbating Coptic-Moslem tensions.

As the emotional intensity surrounding these questions has mounted, people here invariably draw parallels with Iran and Lebanon, usually denying before they are ever asked that Egypt could face similar violence and unrest in the name of religion.

But the fundamentalists clearly feel they have the initiative and momentum and that nothing should be allowed to stand in their way.

The Moslem Brotherhood, or Ikhwan, perhaps the best organized and in many ways the most moderate of the fundamentalist groups, has a powerful political voice. Banned as a party since 1954, it has made major inroads in the political system recently by working inside the New Wafd and other parties. It supports the current government policy of "cleansing" the laws, but it disagrees over long-range goals.

"The government doesn't want sharia but wants power forever," said newspaper columnist and Ikhwan member Mohammed Abdul Qadoos. "Of course that will lead to a conflict in the end."

Salama, a religious leader from Suez, became something of a hero after he rallied his followers to fight the Israelis there in the 1973 war. His followers have said they might march Friday with copies of the Koran held before them as protection, as earlier Moslem fighters once protected themselves with pages of the book on their spears.

But Salama insisted that "the famous wrong idea about the Islamic groups in Egypt is that they use violence and violent activity, even though in all their movements they always work in the framework of the laws and legal structure."

A wiry little man, his thin face framed by a white beard, Salama wore a red Fez, a white double-breasted blazer, blue rubber sandals and a beatific smile as he prayed at the mosque he has been building for 11 years. It stands now as a concrete, brick and iron-rod symbol of other incomplete goals and of his pressure to obtain them.

"We thank God," he said, "there are not in Egypt what we call extremists or terrorists. These names don't exist among Islamic groups."