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Israel has four main demands: and end to Gaza rocket fire, a halt to Palestinian attacks, international supervision of a truce and an agreement to stop Hamas from re-arming. Hamas demands a cessation of Israeli attacks and opening of vital Gaza-Israel cargo crossings, Gaza's main lifeline.
U.S. officials maintained their firm support for Israel and squarely blamed Hamas.
Vice President Dick Cheney said Israel "didn't seek clearance or approval from us" before pushing into Gaza.
Sens. Harry Reid and Dick Durbin — the top two Democrats in the chamber — and Senate Republican leader Mitch McConnell all described Israel's actions as understandable.
"I think what the Israelis are doing is very important," Reid said. "I think this terrorist organization, Hamas, has got to be put away. They've got to come to their senses."
Israeli President Shimon Peres said that Israel had to push forward and that a cease-fire was pointless without a halt to Hamas rocket fire.
"Well, clearly, if there is somebody (who) can stop terror with a different strategy, we shall accept it," he said on ABC's "This Week."
"We shall not accept the idea that Hamas will continue to fire and we shall declare a cease-fire. It does not make any sense."
Palestinians said the Israeli military broke into broadcasts on the Hamas TV channel, Al Aqsa, appealing to Palestinians not to agree to serve as human shields for the militants. The message read, "Israel is acting only against Hamas and has no interest in harming you."
The ground operation is the second phase in an offensive that began as a weeklong aerial onslaught aimed at halting Hamas rocket fire that has reached deeper and deeper into Israel, threatening major cities and one-eighth of Israel's population of 7 million.
More than 45 rockets and mortar shells fell in Israel on Sunday morning, sending residents scrambling for bomb shelters. Four Israelis were lightly wounded.
In Gaza City, civilians cowered inside as battles raged, while terrified residents in other areas fled in fear. In the southern town of Rafah, one man loaded a donkey cart with mattresses and blankets preparing to flee.
Lubna Karam, 28, said she and the other nine members of her family spent the night huddled in the hallway of their Gaza City home. The windows of the house were blown out days earlier in an Israeli airstrike, and the family has been without electricity for a week, surviving without heat and eating cold food.