I have to prove that, if two curves $C_1$ and $C_2$ have the same tangent line at point p then: $\DeclareMathOperator{\mult}{mult}$$\DeclareMathOperator{\ord}{ord}$
$$\mult_p(C_1 \cap C_2) > \ord_p(C_1)\ord_p(C_2).$$
In a lecture we proved that $\mult_p(C_1 \cap C_2)\geq\ord_p(C_1)$ ord$_p$$(C_2)$ and only mentioned that the equality holds when there isn’t a tangent line at point $p$, shared by both curves. Now my idea was to assume we have such tangent $L$ line and use definition of tangent line that $\mult_p(L\cap C_1)> \ord_p(C_1)$ and $\mult_p(L \cap C_2) > \ord_p(C_2)$. From here follows:
$$\mult_p(L \cap C_1)\cdot\mult_p(L\cap C_2)>\ord_p(C_1)\cdot\ord_p(C_2).$$
Now in general I assume $\mult_p(L \cap C_1)\cdot\mult_p(L \cap C_2)= \mult_p(C_1 \cap C_2)$ is not true, so I am wondering whether there is some connection between $\mult_p(L \cap C_1)\cdot\mult_p(L \cap C_2)$ and $\mult_p(C_1 \cap C_2)$ such that I could use in this case prove inequality or any other hint?