Sentence:  This is applied in Chapter 9 to prove the first congruence <WSD>subgroup</WSD> theorem, which asserts that <WSD>g</WSD>."z" = "z" for all "z" in the center of the Drinfel'd double "D"("H") and all <WSD>g</WSD> in the principal congruence <WSD>subgroup</WSD>. 
Meaning of the word marked with <WSD> tags:  A subset "H" of a group "G" that is itself a group and has the same binary operation as "G".