Equal Surfaces do not Intersect
Let $S_1$ and $S_2$ be equal surfaces in $R$ upon which the value of $F$ on $S_1$ is different from the value of $F$ on $S_2$.
Then $S_1$ and $S_2$ do not intersect.
- $\forall p \in S_1: \map F p = C_1$
- $\forall p \in S_2: \map F p = C_2$
By hypothesis, $C_1 \ne C_2$.
Then $\map F p = C_1$ and also $\map F p = C_2$.
Hence the result by Proof by Contradiction.