In order to use ${\displaystyle \prod _{j=1}^{m}h_{j}(\Psi _{j})}$ as a template to further factorize ${\displaystyle f(\Theta )}$, all variables outside of ${\displaystyle \Theta }$ need to be fixed. To this end, let ${\displaystyle {\bar {\theta }}}$ be an arbitrary fixed assignment to the variables from ${\displaystyle U\setminus \Theta }$ (the variables not in ${\displaystyle \Theta }$). For an arbitrary set of variables ${\displaystyle X}$, let ${\displaystyle {\bar {\theta }}[X]}$ denote the assignment ${\displaystyle {\bar {\theta }}}$ restricted to the variables from ${\displaystyle X\setminus \Theta }$ (the variables from ${\displaystyle X}$, excluding the variables from ${\displaystyle \Theta }$).

Moreover, to factorize only ${\displaystyle f(\Theta )}$, the other factors ${\displaystyle g_{1}(\Phi _{1}),g_{2}(\Phi _{2}),...,g_{n}(\Phi _{n})}$ need to be rendered moot for the variables from ${\displaystyle \Theta }$. To do this, the factorization

${\displaystyle \Pr(U)=f(\Theta )\prod _{i=1}^{n}g_{i}(\Phi _{i})}$

will be re-expressed as

${\displaystyle \Pr(U)={\bigg (}f(\Theta )\prod _{i=1}^{n}g_{i}(\Phi _{i}\cap \Theta ,{\bar {\theta }}[\Phi _{i}]){\bigg )}{\bigg (}\prod _{i=1}^{n}{\frac {g_{i}(\Phi _{i})}{g_{i}(\Phi _{i}\cap \Theta ,{\bar {\theta }}[\Phi _{i}])}}{\bigg )}}$

For each ${\displaystyle i=1,2,...,n}$: ${\displaystyle g_{i}(\Phi _{i}\cap \Theta ,{\bar {\theta }}[\Phi _{i}])}$ is ${\displaystyle g_{i}(\Phi _{i})}$ where all variables outside of ${\displaystyle \Theta }$ have been fixed to the values prescribed by ${\displaystyle {\bar {\theta }}}$.

Let ${\displaystyle f'(\Theta )=f(\Theta )\prod _{i=1}^{n}g_{i}(\Phi _{i}\cap \Theta ,{\bar {\theta }}[\Phi _{i}])}$ and ${\displaystyle g'_{i}(\Phi _{i})={\frac {g_{i}(\Phi _{i})}{g_{i}(\Phi _{i}\cap \Theta ,{\bar {\theta }}[\Phi _{i}])}}}$ for each ${\displaystyle i=1,2,\dots ,n}$ so

${\displaystyle \Pr(U)=f'(\Theta )\prod _{i=1}^{n}g'_{i}(\Phi _{i})=\prod _{j=1}^{m}h_{j}(\Psi _{j})}$

What is most important is that ${\displaystyle g'_{i}(\Phi _{i})={\frac {g_{i}(\Phi _{i})}{g_{i}(\Phi _{i}\cap \Theta ,{\bar {\theta }}[\Phi _{i}])}}=1}$ when the values assigned to ${\displaystyle \Phi _{i}}$ do not conflict with the values prescribed by ${\displaystyle {\bar {\theta }}}$, making ${\displaystyle g'_{i}(\Phi _{i})}$ "disappear" when all variables not in ${\displaystyle \Theta }$ are fixed to the values from ${\displaystyle {\bar {\theta }}}$.

Fixing all variables not in ${\displaystyle \Theta }$ to the values from ${\displaystyle {\bar {\theta }}}$ gives

${\displaystyle \Pr(\Theta ,{\bar {\theta }})=f'(\Theta )\prod _{i=1}^{n}g'_{i}(\Phi _{i}\cap \Theta ,{\bar {\theta }}[\Phi _{i}])=\prod _{j=1}^{m}h_{j}(\Psi _{j}\cap \Theta ,{\bar {\theta }}[\Psi _{j}])}$

Since ${\displaystyle g'_{i}(\Phi _{i}\cap \Theta ,{\bar {\theta }}[\Phi _{i}])=1}$,

${\displaystyle f'(\Theta )=\prod _{j=1}^{m}h_{j}(\Psi _{j}\cap \Theta ,{\bar {\theta }}[\Psi _{j}])}$

Letting ${\displaystyle h'_{j}(\Theta \cap \Psi _{j})=h_{j}(\Psi _{j}\cap \Theta ,{\bar {\theta }}[\Psi _{j}])}$ gives:

${\displaystyle f'(\Theta )=\prod _{j=1}^{m}h'_{j}(\Theta \cap \Psi _{j})}$ which finally gives:

${\displaystyle \Pr(U)={\bigg (}\prod _{j=1}^{m}h'_{j}(\Theta \cap \Psi _{j}){\bigg )}{\bigg (}\prod _{i=1}^{n}g'_{i}(\Phi _{i}){\bigg )}}$