The first statement is elementary. To prove the second, we must make sure to handle the “multiplicities” $b_{\sigma}$ and $b_{\sigma^{\prime}}$ correctly. Parametrize the interior of $\sigma$ by coordinates $(w_{1},\dots,w_{p})$ using $w_{0}=b_{0}^{-1}(1-\sum_{1}^{p}b_{i}w_{i})$. By Remark 1.3 we have

$$b_{\sigma}\lambda_{\sigma}=|dw_{1}\wedge\dots\wedge dw_{p}|$$

Similarly, we parametrize the interiors of $\sigma^{\prime}$ and $\sigma_{t}$ using coordinates $(w_{1},\dots,w_{q})$ and $(w_{1},\dots,w_{q},x^{\prime\prime}_{q+1},\dots,x^{\prime\prime}_{p})$, respectively. Then

$$b_{\sigma^{\prime}}\lambda_{\sigma^{\prime}}=|dw_{1}\wedge\dots\wedge dw_{q}|.$$

The required formula now follows from an elementary computation. ∎