(\bfone + W) (\bfone - W)\inv
 \cr
&=& - (\bfone + J_0 W J_0\inv) (\bfone - J_0 W J_0\inv)\inv
(\bfone + W) (\bfone - W)\inv
 \cr
&=& - (\bfone - W ) (\bfone + W )\inv
(\bfone + W) (\bfone - W)\inv \qquad  =- \bfone 
\endmatrix$$

[*Step 6. Let $W\in \calw$ and $J= \sfK(W)$. Then $J+ J_0$ is invertible and $J \in \jj_\omega$*]{}. In fact, $J + J_0 =  J_0\left( \bfone + {\bfone + 
W \over \bfone - W} \right) = {2 J_0 \over \bfone - W}$ is invertible. Now, repeating argumantations from [*Step 4*]{} the positivity of $\bfone - 
W^\sft W$ is equivalent to the positivity of $J_0 J^\sft - JJ_0$, which means the tameness of $J$.

The lemma is proved.

Remark. One can consider $\sfK:\calw \to \jj_\omega \subset \jj$ a generalized Cayley transformation defined on a bounded domain $\calw$ in the complex linear space $\barr \endo_\cc(V,J_0)$ of $J_0$-antilinear endomorphisms of $V$.

[1.3. Almost-Complex Submanifolds, Complex Curves, Energy and Area.]{}

Let $J$ be some linear complex structure on $\rr^{2n}$, , $J\in
\endo(\rr^{2n}), J^2=-\id $. Let $h$ be some $J$-Hermitian metric on $\rr^{2n}$, $\omega =\omega_h$ its associated $(1,1)$-form and $g=g_h$ corresponding Riemannian metric. Put $\sigma_k =  {1\over k!}\omega^k$.

Wirtinger Inequality. 