How is Mary asked to prove that the girls were lying?
The part you are asking about is in Act 3 where John Proctor brings Mary Warren to court to prove the girls are lying. She testifies that they faked fainting. When she does this, Parris and Hathorne challenge her to prove it by fainting on the spot. They say that if she faked it before, she should be able to fake it again. They say that when she fainted before, she was all cold and they want her to become like that again so that they can see that she is telling the truth and that the other girls are lying.