=_{\tau\to\sigma} f_2)$$ $$e_b^P: \forall \varphi_1, \varphi_2 : o \,.\, \left((\varphi_1 \supset
\varphi_2) \wedge (\varphi_2 \supset \varphi_1)\right) \supset
(\varphi_1 =_{o} \varphi_2)$$

For an arbitrary set of formulas $\Delta$ we write $\Delta {\ensuremath{\vdash}}_S
\varphi$ if $\varphi$ is derivable from a subset of $\Delta$ in system $S$.

We now define a mapping ${\ensuremath{\lceil-\rceil}}$ from types and terms of CPRED$\omega$ to terms of ${\ensuremath{{\cal I}}}_s$, and a mapping $\Gamma(-)$ from sets of terms of CPRED$\omega$ to sets of terms of ${\ensuremath{{\cal I}}}_s$ providing necessary context. We assume that ${\ensuremath{{\cal B}}}\subseteq \Sigma_s$ and $\Sigma_\tau \subseteq \Sigma_s$ for $\tau \in {\ensuremath{{\cal T}}}$, i.e. that all base types and all constants of CPRED$\omega$ occur as constants in ${\ensuremath{{\cal I}}}_s$, and also $V_\tau \subseteq V_s$ for $\tau \in {\ensuremath{{\cal T}}}$. The definition of ${\ensuremath{\lceil-\rceil}}$ is inductive:

-   ${\ensuremath{\lceil\tau\rceil}} = \tau$ for $\tau \in {\ensuremath{{\cal B}}}$,

-   ${\ensuremath{\lceilo\rceil}} = {\ensuremath{\mathrm{Prop}}}$,

-   ${\ensuremath{\lceil\tau_1\to\tau_2\rceil}} = {\ensuremath{\lceil\tau_1\rceil}} \to {\ensuremath{\lceil\tau_2\rceil}}$ for $\tau_1,\tau_2 \in {\ensuremath{{\cal T}}}$,

-   ${\ensuremath{\lceilc\rceil}} = c$ if $c \in \Sigma_\tau$ for some $\tau \in {\ensuremath{{\cal T}}}$,

-   ${\ensuremath{\lceilx\rceil}} = x$ if $x \in V_\tau$ for some $\tau \in {\ensuremath{{\cal T}}}$,

-   ${\ensuremath{\lceilt_1 t_2\rceil}} = {\ensuremath{\lceilt_1\rceil}} {\ensuremath{\lceilt_2\rceil}}$,

-   ${\ensuremath{\lceil\lambda x : \tau \,.\, t\rceil}} = \lambda x \,.\,
      {\ensuremath{\lceilt\rceil}}$,

-   ${\ensuremath{\lceil\varphi 