Why $|W_t|-L_t$ is a Brownian motion

Question

I have proved Tanaka's formula, $$ |W_t|=|W_0|+\int_0^t \operatorname{sgn}(W_s)dW_s + L_t $$ Here $L_t$ is the local time at $0$ of the Brownian motion $W_t$. But I don't know why it implies that $|W_t|-L_t$ is a Brownian motion a.s. I think it is because of an stochastic equations argument, but I am stuck