When a person, having taken an oath before a competent tribunal, officer, or person, in any case in which a law of the U.S. authorizes an oath to be administered, that he will testify, declare, depose, or certify truly, or that any written testimony, declaration, deposition, or certificate by him subscribed, is true, willfully and contrary to such oath states or subscribes any material matter which he does not believe to be true; or in any declaration, certificate, verification, or statement under penalty of perjury, willfully subscribes as true any material matter which he does not believe to be true; 18 USC
In order for a person to be found guilty of perjury the government must prove: the person testified under oath before [e.g., the grand jury]; at least one particular statement was false; and the person knew at the time the testimony was false.
The testimony of one witness is not enough to support a finding that the testimony was false. There must be additional evidence, either the testimony of another person or other evidence, which tends to support the testimony of falsity. The other evidence, standing alone, need not convince that the testimony was false, but all the evidence on the subject must do so.