Detailed proof of a theorem on coincidence of homological dimensions of Fr\'echet algebras of smooth functions on a manifold with the dimension of the manifold

Given work contains the full text of the proof of the following assertion: For the topological algebra $C^{\mathrm{\infty }}(\mathcal{M})$ of smooth functions on a smooth $m$-dimensional real manifold $\mathcal{M}$ the small global dimension $(C^{\mathrm{\infty }}(\mathcal{M}))$, the global homological dimension $(C^{\mathrm{\infty }}(\mathcal{M}))$ and the bidimension $(C^{\mathrm{\infty }}(\mathcal{M}))$ are equal to $m$ (all dimensions are understood in the sense of the homology of topological (locally convex) algebras).