$d_{\mathbb{H}}(p,q)=|\log(pq;rs)|$.

Question

I'd like some assistance in demonstrating $d_{\mathbb{H}}(p,q)=|\log(pq;rs)|$ where $\mathbb{H}$ is a Poincare Upper Half-Plane, and $(pq;rs)$ is a cross ratio of $p,q,r,s$. ($p,q$ are on a geodesic in $\mathbb{H}$ and $r,s$ are the intersection of $x$-axis and the geodesic.)

I tried as follows.

For some $f \in PSL(2,\mathbb{R})$, $p,q,r,s$ moves to $0,i,\infty, \lambda i$ for some $\lambda >0$.

$f$ is an isometry map, so $d_{\mathbb{H}}(p,q)=d_{\mathbb{H}}(i,\lambda i)$.

And, I moved $i,\lambda i$ back to the Poincare disk model $D$ and moved to $B$, the Beltrami model.

By a conformal map that maps $\mathbb{H}$ to $D$, $0 \mapsto -1, \infty \mapsto 1, i \mapsto 0=O, \lambda i \mapsto \frac{\lambda -1}{\lambda +1}=q'$.

And these points get mapped to $-1,1,O,q' \in B$

$$d_{\mathbb{H}}(i,\lambda i)=d_D(O,q')=d_B(O,q')=\frac{1}{2}\left| \log(Oq';(-1) (1))\right|=\frac{1}{2}\log\frac{1\cdot \left\| 1-q'\right\|}{\left\|q'+1 \right\|\cdot 1} $$

And, $\left\|q'+1 \right\|=\frac{\lambda -1}{\lambda +1}+1, \left\| 1-q'\right\|= 1- \frac{\lambda -1}{\lambda +1}$, so $\frac{\left\| 1-q'\right\|}{\left\|q'+1 \right\|}=\frac{1}{\lambda}$.

Therefore, $\frac{1}{2}\log\frac{1\cdot \left\| 1-q'\right\|}{\left\|q'+1 \right\|\cdot 1}=\frac{1}{2}\log\frac{1}{\lambda}=d_{\mathbb{H}}(p,q)$.

$(pq;rs)=((i) (\lambda i);0 \infty)=\frac{i(\lambda i- \infty)}{\lambda i(i-\infty)}=\frac{1}{\lambda}$ because cross-ratio is preserved by $f$.

However, this is different from my $d_\mathbb{H}(p,q)$ because there is $\frac{1}{2}$.

I have no clue where I made this mistake. Maybe it's because of the equation $d_D(O,q')=d_B(O,q')$, but why is this wrong?