The splitting of the de Rham cohomology of soft function algebras is multiplicative

Let $A$ be a real soft function algebra. In arXiv:2208.11431 we have obtained a canonical splitting ${\mathrm{H}}^{\ast }({\mathrm{\Omega }}_{A\mathrm{\mid }\mathrm{R}}^{\bullet })\cong {\mathrm{H}}^{\ast }(X,\mathrm{R})\oplus \text{(something)}$ via the canonical maps ${\mathrm{\Lambda }}_A:{\mathrm{H}}^{\ast }(X,\mathrm{R})\to {\mathrm{H}}^{\ast }({\mathrm{\Omega }}_{A\mathrm{\mid }\mathrm{R}}^{\bullet })$ and ${\mathrm{\Psi }}_A:{\mathrm{H}}^{\ast }({\mathrm{\Omega }}_{A\mathrm{\mid }\mathrm{R}}^{\bullet })\to {\mathrm{H}}^{\ast }(X,\mathrm{R})$. In this paper we prove that these maps are multiplicative.