ii) follows from symmetry and multilinearity of the intersection product ([Ful98, Proposition 2.5]). For iii) we reduce first to the case where $\mathfrak{X}^{\prime}$ and $\mathfrak{X}$ have reduced special fibre. Let $\mathfrak{Y}^{\prime}$ respectively $\mathfrak{Y}$ be the canonical formal models with reduced special fibre as in 4. This construction is functorial and we obtain a commutative diagram

Assuming that we know the claim for reduced special fibres we obtain

	$\displaystyle\Deg(\varphi^{\textup{an}})c_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})$	$\displaystyle=\Deg(\varphi^{\prime an})(\iota^{\textup{an}})_{\ast}(c_{1}(\iota^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\iota^{\ast}\mathfrak{L}_{n}))$
		$\displaystyle=(\iota^{\textup{an}})_{\ast}(\varphi^{\prime an})_{\ast}(c_{1}(\varphi^{\prime\ast}\iota^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\varphi^{\prime\ast}\iota^{\ast}\mathfrak{L}_{n}))$
		$\displaystyle=(\varphi^{\textup{an}})_{\ast}(\iota^{\prime an})_{\ast}(c_{1}(\iota^{\prime\ast}\varphi^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\iota^{\prime\ast}\varphi^{\ast}\mathfrak{L}_{n}))$
		$\displaystyle=(\varphi^{\textup{an}})_{\ast}(c_{1}(\varphi^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\varphi^{\ast}\mathfrak{L}_{n})).$

So from now on assume that $\mathfrak{X}^{\prime}$ and $\mathfrak{X}$ have reduced special fibre. Let $Y$ be an irreducible component of $\tilde{\mathfrak{X}}$ with corresponding Shilov point $\zeta_{Y}$. Let $\zeta_{1},...,\zeta_{r}$ be the preimages of $\zeta_{Y}$ under $\varphi^{\textup{an}}$ with corresponding irreducible components $Y_{1},...,Y_{r}$ of $\tilde{\mathfrak{X}^{\prime}}$. If $Y$ is proper then clearly all the $Y_{i}$ are proper. If on the other hand one of the $Y_{i}$ is proper then $Y$ is proper by [GW10, Proposition 12.59]. In this case we can use the projection formula to calculate:

	$\displaystyle(\varphi^{\textup{an}})_{\ast}\left(c_{1}(\varphi^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\varphi^{\ast}\mathfrak{L}_{n})\right)(\zeta_{Y})$	$\displaystyle=\sum_{i=1}^{r}c_{1}(\varphi^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\varphi^{\ast}\mathfrak{L}_{n})(\zeta_{i})$
		$\displaystyle=\sum_{i=1}^{r}\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(\tilde{\varphi}_{\ast}Y_{i})$
		$\displaystyle=\sum_{i=1}^{r}\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(Y)\cdot[\tilde{K}(Y_{i}):\tilde{K}(Y)]$

As already mentioned in Definition 4.2, $\varphi^{\textup{an}}$ is finite outside a lower dimensional analytic subset. Hence we may apply equation (3) in the proof of [Gub98, Proposition 4.5] to see that the last term in the display equals $\Deg(\varphi^{\textup{an}})\cdot c_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})(\zeta_{Y})$.
On the other hand, if $Y$ is an irreducible component of $\tilde{\mathfrak{X}}^{\prime}$ whose image is not an irreducible component of $\tilde{\mathfrak{X}}$ then its degree with respect to the line bundles $\varphi^{\ast}\mathfrak{L}_{1},...,\varphi^{\ast}\mathfrak{L}_{n}$ is $0$ by the projection formula, as the image is of lower dimension. This proves iii).
For i) let $\mathfrak{X}^{\textup{an}}=\sum_{j}m_{j}X_{j}$ be the decomposition into prime cycles. It is then enough to prove the claim for each $X_{j}$ and by definition of the measure we may hence assume that $\mathfrak{X}$ has irreducible and reduced generic fibre and reduced special fibre. Let $S$ be the set of all $\zeta_{Y}$ where $Y$ is a proper irreducible component of $\tilde{\mathfrak{X}}$ with $\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(Y)\neq 0$. Then $S$ is discrete as $\red^{-1}(Y)$ is an open neighbourhood of $\zeta_{Y}$ which does not contain any other points of $S$. Furthermore $X$ is the union of all $\red^{-1}(Y)$ where $Y$ runs over all irreducible components of $\tilde{\mathfrak{X}}$ and as all of these sets contain at most one point of $S$ and by paracompactness of $X$, every $x\notin S$ has an open neighbourhood which does not intersect $S$ and hence $S$ is closed. By definition $c_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})$ is of the desired form and its support is contained in the relative interior of $X$ over $K$ by Corollary A.4. ∎