Let $S_1$, $S_2$ be two regular surfaces such that they touch each other at $p\in S_1\cap S_2$. How can we show that the tangent planes $T_pS_1$ and $T_pS_2$ are identical? Thank you for your help.
Edit: At the beginning we have $S_1 \cap S_2 = \emptyset$. We then apply a rigid motion (translation) in $S_1$ towards $S_2$, i.e. we move $S_1$ towards $S_2$ and stop immediately when $S_1\cap S_2\neq \emptyset.$ In this case there is $p\in S_1\cap S_2$ and the tangent planes should be identical.