Heineman's win sends him to the November general election against Omaha investment banker Mark Lakers, who ran unopposed in the Democratic primary.
The Omaha World-Herald said that with more than half of the precincts reporting, Heineman was the victor with 90 percent of the vote to about 5 percent each for Paul Anderson and Christopher Geary, both of Omaha.
Heineman became governor in January 2005 when Mike Johanns resigned to become U.S. agriculture secretary in President George W. Bush's administration and then was elected to a full term in 2006.
In the 1st District, Fortenberry had 84 percent of the vote to 10 percent for David Hunt and 6 percent for Ralph Bodie with more than two-thirds of the votes counted.
Ivy Harper and Jessica Lynn Turek were neck-and-neck with 37 percent each in the four-candidate Democratic field.
Terry, running for a seventh term in the 2nd District, had 65 percent of the vote to 35 percent for Matt Sakalosky. Terry will face state Sen. Tom White, who was unopposed in the Democratic primary, in November.
Smith had 88 percent of the vote to 12 percent for Dennis Parker in the sprawling, 69-county 3rd District. Smith will go against Democrat Rebekah Davis in the general election. Davis was unopposed in the primary.
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