[proofplan] We rewrite the displayed summation hypothesis as an identity in the incidence algebra, where the zeta element encodes interval summation. The Möbius element is the convolution inverse of the zeta element, so right multiplication by the Möbius element cancels the zeta element. Local finiteness ensures that all interval sums are finite and that the rearrangements needed for associativity are valid. Evaluating the resulting identity on an arbitrary interval then gives the stated inversion formula. [/proofplan] [step:Define the zeta, delta, and Möbius elements in the incidence algebra] Let $\zeta_P\in I(P;R)$ denote the $R$-valued zeta element defined by \begin{align*} \zeta_P(u,v)=1_R \end{align*} for every $u,v\in P$ with $u\le v$. Let $\delta_P\in I(P;R)$ denote the delta element defined by $\delta_P(u,u)=1_R$ and $\delta_P(u,v)=0_R$ when $u<v$. Let $\mu_P^{\mathbb Z}$ be the integer-valued Möbius function of $P$. Define the $R$-valued Möbius element, still denoted $\mu_P\in I(P;R)$, by \begin{align*} \mu_P(u,v)=\mu_P^{\mathbb Z}(u,v)1_R. \end{align*} By the defining property of the Möbius function, for every $u,v\in P$ with $u\le v$ we have \begin{align*} \sum_{u\le w\le v}\mu_P(w,v)=\delta_P(u,v). \end{align*} Equivalently, this element satisfies \begin{align*} \zeta_P*\mu_P=\delta_P. \end{align*} [/step] [step:Rewrite the hypothesis as right multiplication by the zeta element] For fixed $x,y\in P$ with $x\le y$, the convolution convention gives \begin{align*} (F*\zeta_P)(x,y)=\sum_{x\le z\le y}F(x,z)\zeta_P(z,y). \end{align*} Since $\zeta_P(z,y)=1_R$ for each $z$ with $x\le z\le y$, this becomes \begin{align*} (F*\zeta_P)(x,y)=\sum_{x\le z\le y}F(x,z). \end{align*} The hypothesis therefore says that $G(x,y)=(F*\zeta_P)(x,y)$ for every comparable pair $x\le y$, hence \begin{align*} G=F*\zeta_P \end{align*} in $I(P;R)$. [guided] The summation in the theorem is not an arbitrary sum; it is exactly the convolution of $F$ with the zeta element on the right. To see this, fix $x,y\in P$ with $x\le y$. By definition of convolution in $I(P;R)$, \begin{align*} (F*\zeta_P)(x,y)=\sum_{x\le z\le y}F(x,z)\zeta_P(z,y). \end{align*} The element $\zeta_P$ has value $1_R$ on every comparable pair, so for each index $z$ in the finite interval $[x,y]$ we have $\zeta_P(z,y)=1_R$. Therefore \begin{align*} (F*\zeta_P)(x,y)=\sum_{x\le z\le y}F(x,z)1_R=\sum_{x\le z\le y}F(x,z). \end{align*} The displayed hypothesis is precisely the assertion that this last expression equals $G(x,y)$ for every comparable pair $x\le y$. Since elements of the incidence ring are functions on comparable pairs, equality at every such pair gives \begin{align*} G=F*\zeta_P. \end{align*} This is the algebraic form of the interval summation relation. [/guided] [/step] [step:Cancel the zeta element by multiplying on the right by the Möbius element] Right multiply the identity $G=F*\zeta_P$ by $\mu_P$. For each interval $[x,y]$, local finiteness makes every convolution sum finite, so convolution is associative by finite rearrangement of sums. Hence \begin{align*} G*\mu_P=(F*\zeta_P)*\mu_P=F*(\zeta_P*\mu_P). \end{align*} Using $\zeta_P*\mu_P=\delta_P$, we get \begin{align*} G*\mu_P=F*\delta_P. \end{align*} The element $\delta_P$ is the right identity for convolution: for every $x\le y$, \begin{align*} (F*\delta_P)(x,y)=\sum_{x\le z\le y}F(x,z)\delta_P(z,y)=F(x,y), \end{align*} because $\delta_P(z,y)=0_R$ unless $z=y$, and $\delta_P(y,y)=1_R$. Thus \begin{align*} G*\mu_P=F. \end{align*} [/step] [step:Evaluate the convolution identity to obtain the inversion formula] Let $x,y\in P$ with $x\le y$. Evaluating $G*\mu_P=F$ at $(x,y)$ gives \begin{align*} F(x,y)=(G*\mu_P)(x,y). \end{align*} Expanding the convolution on the right-hand side yields \begin{align*} (G*\mu_P)(x,y)=\sum_{x\le z\le y}G(x,z)\mu_P(z,y). \end{align*} Therefore \begin{align*} F(x,y)=\sum_{x\le z\le y}G(x,z)\mu_P(z,y). \end{align*} Since $x,y$ were arbitrary comparable elements of $P$, the formula holds on every interval of $P$. [/step]