Conditional inequalities and independence

Let $A,B,C$ be three events such that $P(C)P(C^{c})>0$. Consider the following statements. $S1$: $P(A\mid C)>P(B\mid C)$ and $P(A\mid C^{c})>P(B\mid C^{c})$ together imply that $P(A)>P(B)$. $S2$: $P(A\mid C)=P(A\mid C^{c})$ implies that $A$ and $C$ are independent. Then