lemma, we note some properties of a good pair.

\[lem-gp\] Let $(\phi,\Omega)$ be a good pair for $A$. Then:

1.  $\nu_{(\phi,\Omega)}^2=1$.

2.  Setting $H'=({\mathrm{id}}_A \otimes \mu)(\Delta \otimes {\mathrm{id}}_A) {\colon}A \otimes A \to A \otimes A$, we have: $$(\phi \otimes \phi)H'=\phi \otimes \phi {\quad \text{and} \quad}H'(\Omega \otimes \Omega)=\Omega \otimes \Omega.$$ Graphically: $${\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-2.eps}}}} \,=\;\, {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-2b.eps}}}} {\quad \text{and} \quad}{\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-2c.eps}}}} \,=\;\, {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-2d.eps}}}}\;\,.$$

3.  For any $\epsilon \in\{0,1\}$, $$({\mathrm{id}}_A \otimes fS^\epsilon \otimes {\mathrm{id}}_A)\Delta_3 \Omega=({\mathrm{id}}_A \otimes S^\epsilon \otimes {\mathrm{id}}_A)\Delta_3 \Omega$$ where $f$ is as in Axiom [(GP4)]{}. Graphically: $$\psfrag{h}[Bc][Bc]{\scalebox{.9}{$f$}}
    {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{gp-ppt-3n.eps}}}} \;=\; {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{gp-ppt-4n.eps}}}}
    \, {\quad \text{and} \quad}\;
    {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{gp-ppt-1.eps}}}} \;=\; {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{gp-ppt-2.eps}}}}\,.$$

4.  For any $\epsilon \in\{0,1\}$, $$\phi \mu_3({\mathrm{id}}_A \otimes S^\epsilon h \otimes {\mathrm{id}}_A)=\phi \mu_3({\mathrm{id}}_A \otimes S^\epsilon \otimes {\mathrm{id}}_A)$$ where $h$ is as in Axiom [(GP5)]{}. Graphically: $$\psfrag{h}[Bc][Bc]{\scalebox{.9}{$h$}}
    {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-ppt-7n.eps}}}} \;=\; {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-ppt-8n.eps}}}}
    \, {\quad \text{and} \quad}\;
    {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-ppt-5.eps}}}} \;=\; {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-ppt-6.eps}}}}\,.$$

Let us prove Part (a). We have: $$1 \overset{(i)}{=} {\left\langle\phi \Omega\right\rangle}_I \overset{(ii)}{=} {\left\langle\phi S^2 \Omega\right\rangle}_I \overset{(iii)}{=} \nu_{(\phi,\Omega)}  {\left\langle\phi S \Omega\right\rangle}_I \overset{(iv)}{=} \nu_{(\phi,\Omega)}^2 {\left\langle\phi \Omega\right\rangle}_I  \overset{(v)}{=} \nu_{(\phi,\Omega)}^2.$$ Here $(i)$ and $(v)$ follow from (GP1), $(ii)$ from the involutivity of $S$, and $(iii)$ and $(iv)$ from (GP3). Part (b) follows from Axioms (GP2)-(GP3), Part (a), and the fact that $$H'=\tau_{A,A}(S \otimes S) H (S \otimes S) \tau_{A,A}.$$ Let us prove Part (c) for 