property of distribution function

Question

Let $f$ a continuous map from $\mathbb{R} \rightarrow \mathbb{R}$ and let $L_1, L_2$ 2 probability measures on $\mathbb{R}$. Let $K$ be a closed set in $\mathbb{R}$. In a proof, I want to use the following inequality: $| \int fI_{K^c} dL_1 - \int fI_{K^c} dL_2| \leq |\int (f \wedge 1)^+ dL_1 - (f \wedge 1)^+ dL_2| $ with $I$ the indicator function, $f \wedge g = \min(f,g)$ and $f^+ = fI_{f \geq 0}$ . Can someone prove this ? (or give me a counterexample) EDIT: you do not answer this, I have found a mistake in another part of the proof