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The military has been unable to quell the demonstrations and a civil disobedience campaign of strikes against the coup and the detention of elected leader Aung San Suu Kyi and others, even with a promise of new elections and stern warnings against dissent.
Tens of thousands of people massed peacefully in the second city of Mandalay, where Saturday's killings took place, witnesses said.
"They aimed at the heads of unarmed civilians. They aimed at our future," a young protester told the crowd.
The Foreign Ministry said in a statement that despite "unlawful demonstrations, incitements of unrest and violence, the authorities concerned are exercising utmost restraint through minimum use of force to address the disturbances", adding they were maintaining public safety in line with domestic laws and international practices.
In the main city of Yangon, thousands of mostly young people gathered at different sites to chant slogans and sing.
"Us young people have our dreams but this military coup has created so many obstacles," said Ko Pay in Yangon. "That's why we come out to the front of the protests."
In Myitkyina in the north, people laid flowers for the dead protesters. Big crowds marched in the central towns of Monywa and Bagan, in Dawei and Myeik in the south, Myawaddy in the east and Lashio in the northeast, posted pictures showed.
At the tourist spot of Inle Lake, people including Buddhist monks took to a flotilla of boats holding aloft portraits of Suu Kyi and signs saying "military coup - end".
The more than two weeks of protests had been largely peaceful until Saturday, unlike previous episodes of opposition during nearly half a century of direct military rule to 2011.
The violence looked unlikely to end the agitation.
"The number of people will increase ... We won't stop," protester Yin Nyein Hmway said in Yangon.
Several Western countries that have condemned the coup decried the violence against protesters.
US State Department spokesman Ned Price said the United States was "deeply concerned". France, Singapore, Britain and Germany also condemned the violence and UN Secretary-General Antonio Guterres said lethal force was unacceptable.
Sunday's Foreign Ministry statement reiterated the junta's stance that the takeover was constitutional and said remarks by some embassies and foreign countries "are tantamount to flagrant interference in internal affairs of Myanmar".
Military spokesman Zaw Min Tun has not responded to attempts by Reuters to contact him by telephone for comment.
The trouble in Mandalay began with confrontations between the security forces and striking shipyard workers.
Video clips on social media showed members of the security forces firing at protesters and witnesses said they found the spent cartridges of live rounds.
UN Special Rapporteur for Myanmar Tom Andrews said he was horrified by the deaths of the two, one of them a teenage boy.
"From water cannons to rubber bullets to tear gas and now hardened troops firing point blank at peaceful protesters. This madness must end, now," he said on Twitter.
The state-run Global New Light of Myanmar newspaper said the strikers sabotaged boats at the city's river port and attacked police with sticks, knives and catapults. Eight policemen and several soldiers were injured, it said.
"Some of the aggressive protesters were also injured due to the security measures conducted by the security force in accordance with the law," the newspaper said without mentioning the deaths.
Suu Kyi's National League for Democracy (NLD) called the violence by security forces in Mandalay a crime against humanity.
In an announcement on state-owned media MRTV late on Sunday, authorities said that by planning a big demonstration on Monday, protesters were inciting anarchy and pushing young people towards a path of confrontation "where they will suffer the loss of life".
FUNERAL FOR PROTESTER
A young woman protester, Mya Thwate Thwate Khaing, became the first death among the demonstrators. She was shot in the head on February 9 in the capital Naypyitaw.
Hundreds of people attended her funeral on Sunday.
Military media said the bullet that killed her did not come from any gun used by police and so must have been fired by an "external weapon".
The army says one policeman has died of injuries sustained in a protest.
The army seized power after alleging fraud in Nov. 8 elections that the NLD swept, detaining Suu Kyi and others. The electoral commission dismissed the fraud complaints.
Facebook deleted the military's main page for repeated violations of its standards "prohibiting incitement of violence and coordinating harm".
A rights group said 569 people have been detained in connection with the coup.