Proposition 5

Let $A$ be a formally smooth linearly topologized $k$-algebra. Every continuous homomorphism of $k$-algebras $\varphi : A \to C/C_m$ admits a continuous lifting to $C$.

For every integer $n > m$, let us denote by $\pi_n : C/C_n \to C/C_{n-1}$ the canonical surjection. Since $C$ is identified with the projective limit of the $C/C_n$, to give a continuous lifting of $\varphi$ to $C$ amounts to the same as giving a family $(\varphi_n)_{n > m}$ of continuous homomorphisms of $k$-algebras $\varphi_n : A \to C/C_n$, satisfying $\pi_n \circ \varphi_n = \varphi_{n-1}$. This reduces us, by induction on $m$, to proving the statement when $C_{m+1} = 0$. The ideal $C_m$ is then square zero (for $2m \geq m+1$), whence the proposition since $A$ is formally smooth.