In the first section of the proof of the primitive existence theorem we are trying to show that $$F\left(z+h\right)-F\left(z\right)=\int_{L_{z,z+h}}f$$ where $\alpha, z,z+h$ are collinear and $\alpha=z+xh$ where $h\neq0$
The proof is, $$F\left(z+h\right)-F\left(z\right)=\int_{L_{\alpha,z+h}}f-\int_{L_{\alpha,z}}f\\ =\int^{1}_{0}f\left(\alpha+\left(z+h-\alpha\right)t\right)\left(z+h-\alpha\right)dt-\int^{1}_{0}f\left(\alpha+\left(z-\alpha\right)t\right)\left(z-\alpha\right)dt\\ =h\left(1-x\right)\int^{1}_{0}f\left(z+xh+\left(h-xh\right)t\right)dt+xh\int^{1}_{0}f\left(z+xh-xht\right)dt\\ =h\int^{1}_{x}f\left(z+hs\right)ds-h\int^{0}_{x}f\left(z+hs\right)ds\\ =h\int^{1}_{0}f\left(z+hs\right)ds\\ =\int_{L_{z,z+h}}f$$
Could someone please explain how to go from the 3rd line of the above proof to the 4th? I'm unsure of what $s$ is or where it comes from.
It's a substitution. Set $s=x+(1-x)t.$ Then $ds=(1-x)dt$ and the limits change from $t\in[0,1]$ to $s\in[x,1]$.