: {\mathbb{R}}^2 \times {\Omega}^s  \to {\mathbb{R}}$ and $ \pi_c : {\mathbb{R}}^2 \times {\Omega}^s \to {\Omega}^s$ on points $ x= ( \tilde{\alpha},\tilde{\omega},\tilde{ c}) \in {\mathbb{R}}^2 \times {\Omega}^s$ as: $$\begin{aligned}
    \pi_{\alpha} (x) &:= \tilde{\alpha} 
    &
    \pi_{\omega} (x) &:= \tilde{\omega}
    &
    \pi_{c} (x) &:= \tilde{c} .\end{aligned}$$ For a fixed integer $M \in {\mathbb{N}}$, define the projection maps $\pi_{M}, \pi_{\infty} : {\Omega}^s \to {\Omega}^s$ by: $$\begin{aligned}
\pi_{M}(c) &:= \sum_{k=1}^M [c]_k e_k &&&
\pi_{\infty}(c) &:= c - \pi_{M} (c).
\label{eq:Galerkin1}\end{aligned}$$ Define the projection maps $\pi_{M}' ,\pi_\infty': {\mathbb{R}}^2 \times {\Omega}^s \to {\mathbb{R}}^2 \times {\Omega}^s$ by: $$\begin{aligned}
\pi_{M}'(c) &:=  (\pi_\alpha(x), \pi_\omega(x), \pi_{M} \circ \pi_c (x)) &&&
\pi_{\infty}'(c) &:= (0,0, \pi_{\infty} \circ \pi_c (x)).
\label{eq:Galerkin2}\end{aligned}$$ For any bounded set $X \subseteq {\mathbb{R}}^2 \times {\Omega}^s$, define: $$|X|_k := \sup_{x \in X} \left| [\pi_{c} (x)]_k \right|.$$ We define for $ F$ its Galerkin projection and remainder $ F_M, F_\infty : {\mathbb{R}}^2 \times {\Omega}^{s} \to {\Omega}^{s-1}$ as follows: $$\begin{aligned}
F_M( x) &:= \pi_M \circ F( \pi_M'(x)),  &
F_\infty(x) &:= F(x) - F_M(x).\end{aligned}$$ By construction $ F = F_M + F_\infty$.

To show that there is a unique SOPS to we need to evaluate $F$ not just on single points but on voluminous subsets of its domain. The central subset of ${\mathbb{R}}^2 