 This follows from the Proper Iteration Lemma, which states that whenever formula_10 is a countable support forcing iteration based on formula_11 and formula_12 is a countable elementary substructure of formula_13 for a sufficiently large regular cardinal formula_14, and formula_15 and formula_16 and formula_17 is formula_18-generic and formula_17 forces "formula_20," then there exists formula_21 such that formula_22 is formula_12-generic and the restriction of formula_22 to formula_25 equals formula_17 and formula_17 forces the restriction of formula_22 to formula_29 to be stronger or equal to formula_30.This version of the Proper Iteration Lemma, in which the name formula_30 is not assumed to be in formula_32, is due to Schlindwein.The Proper Iteration Lemma is proved by a fairly straightforward induction on formula_33, and the Fundamental Theorem of Proper Forcing follows by taking formula_34.