$$
    \begin{array}{ll}
         |\forall y (y \in \dot{v} \Longrightarrow y \in \dot{x} \wedge \psi(y))|&= inf_{t \in dom(y)}(|y \in \dot{x}\wedge \varphi(y)| - |y \in \dot{v}|)
         \\
         \\
         &= \inf_{t \in dom(y)}(|y \in \dot{x}||\varphi(y)| - |y \in \dot{v}|)
         \\
         \\
         &= \inf_t (\sup_k \dot{x}(k)\cdot |y=k| \cdot |\varphi(y)| - \sup_g \dot{v}(g)\cdot |y=g|)
         \\
         \\
         &\geq \inf_t \sup_k |y=k|\cdot(\dot{x}(k)|\varphi(y)| - \dot{v}(k))
         \\
         \\
         &= \inf_t \sup_k |y=k|\cdot(\dot{x}(k)|\varphi(y)| - \dot{x}(k)|\varphi(y)|)
         \\
         \\
         &= \inf_t \sup_k \emptyset = \inf_t \emptyset = 1 
         \\
         \\
         & \text{Assim} \space \inf_t (\sup_k \dot{x}(k)\cdot |y=k| \cdot |\varphi(y)| - \sup_g \dot{v}(g)\cdot |y=g|) = 1
    \end{array}
    $$ <wrap right>$\square$</wrap>