This is the proposition 3 page 124 from Algebraic Curves. An Introduction to Algebraic Geometry first edition by William Fulton. It appears in the paragraph of consequences of Max Noether's theorem.
Proposition : Let $C$ be an irreducible cubic and $C', C''$ cubics. Suppose that $C \cap C' = \{ P_1 , \dots , P_9 \}$ where $P_i$ are simple points on $C$ and suppose $C'' \cap C = \{ P_1, \dots , P_8, Q \}$. Then $P_9 = Q$.
Proof : Let $L$ be a line through $P_9$ which doesn't pass through $Q$; $L \cap C = \{ P_9, R, S \}$. Then $LC'' \cap C = (C' \cap C) \cup \{ Q, R, S \}$, so there is a line $L'$ such that $L' \cap C = \{Q, R, S\}$. But then $L' = L$ and so $P_9 = Q$.
Can someone explain how we can say that $L'$ exist ?