Contiguous simplicial maps have homotopic realizations

Let $x={\sum }_i{\lambda }_i{v}_i$ lie in a simplex $\sigma =\{v_0,\dots ,v_n\}$ of $K$. Contiguity says that the vertices $f(v_i)$ and $g(v_i)$ together span a simplex of $L$, so for each $t\in [0,1]$ the barycentric combination $H(x,t):={\sum }_i(1-t){\lambda }_{i}f(v_i)+{\sum }_{i}t{\lambda }_{i}g(v_i)$ lies in $\mathrm{\mid }L\mathrm{\mid }$.

On each simplex of $K$, the formula in step 1.1 is affine in both $x$ and $t$, so it is continuous there. If a point lies on a common face of two simplices, the same barycentric formula is obtained from either side, so the simplexwise formulas patch to a continuous map $H:\mathrm{\mid }K\mathrm{\mid }\times [0,1]\to \mathrm{\mid }L\mathrm{\mid }$.

At $t=0$ the formula gives $\mathrm{\mid }f\mathrm{\mid }(x)$, and at $t=1$ it gives $\mathrm{\mid }g\mathrm{\mid }(x)$. Thus $H$ is a homotopy from $\mathrm{\mid }f\mathrm{\mid }$ to $\mathrm{\mid }g\mathrm{\mid }$.