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After being introduced as one of his “disciples,” Mr. Cruz paid homage to Mr. DeMint, saying he would not be in the U.S. Senate if it were not for Mr. DeMint, who also is credited for playing a big role in putting Sens. Rand Paul of Kentucky, Marco Rubio of Florida and Mike Lee of Utah in office.

A Princeton debate champion and Harvard Law School graduate, Mr. Cruz has earned notice since arriving in the Senate for his combative style — on display most notably in his fiery opposition to the nominations of then-Sen. John F. Kerry for secretary of state and former Sen. Chuck Hagel to head the Pentagon.