\otimes (s(f) \si(n^*x) \xi).$$ Existence of $t$ follows once we show that $$t(m_1 \otimes f_1 \otimes n_1^*)^*t(m_2 \otimes f_2 \otimes n_2^*) = \pi({\left\langlem_1 \otimes f_1 \otimes n_1^*, m_2 \otimes f_2 \otimes n_2^*\right\rangle}.$$ In this case $t$ will be norm-decreasing on finite sums of $m \otimes f \otimes n^* \in E$ and thus can be extended to the entire of $E$. A straightforward computation reveals that $$\begin{aligned}
{\left\langlet(m_1 \otimes f_1 \otimes n_1^*)x_1 \otimes \xi_1, t(m_2 \otimes f_2 \otimes n_2^*) x_2 \otimes \xi_2\right\rangle}_K & = \\
& \hspace{-5cm} =
{\left\langlem_1 \otimes (s(f_1) \si(n_1^*x_1) \xi_1), m_2 \otimes (s(f_2) \si(n_2^*x_2) \xi_2)\right\rangle}_K \\
& \hspace{-5cm} =
{\left\langles(f_1) \si(n_1^*x_1) \xi_1, \si(m_1^* m_2) s(f_2) \si(n_2^*x_2) \xi_2\right\rangle}_H \\
& \hspace{-5cm} =
{\left\langle\xi_1, \si(x_1^* n_1 {\left\langlef_1, m_1^* m_2 f_2\right\rangle} n_2^* x_2) \xi_2\right\rangle}_H\end{aligned}$$ and on the other hand we have that $$\begin{aligned}
{\left\langlex_1 \otimes \xi_1, \pi({\left\langlem_1 \otimes f_1 \otimes n_1^*, m_2 \otimes f_2 \otimes n_2^*\right\rangle}) (x_2 \otimes \xi_2)\right\rangle}_K & = \\
& \hspace{-5cm} =
{\left\langlex_1 \otimes \xi_1, \pi(n_1 {\left\langlef_1, m_1^* m_2 f_2\right\rangle} n_2^*) (x_2 \otimes \xi_2)\right\rangle}_K \\
& \hspace{-5cm} =
{\left\langlex_1 \otimes \xi_1, (n_1 {\left\langlef_1, m_1^* m_2 f_2\right\rangle} n_2^* x_2) \otimes \xi_2\right\rangle}_K \\
& \hspace{-5cm} =
{\left\langle\xi_1, \si(x_1^* n_1 {\left\langlef_1, m_1^* m_2 f_2\right\rangle} n_2^* x_2) \xi_2\right\rangle}_H .\end{aligned}$$

By construction we 