Show that slope for segmented regression changes at threshold z

I am studying a variation of regression model called segmented regression. For a response variable $$y$$ and covariate $$x$$, it is set that the first $$m$$ values of $$x$$ are less than threshold $$z$$ and the remaining are greater than $$z$$. Null hypothesis sets that given $$x$$, the responses $$y_i$$ follow a simple linear model $$\alpha+\beta x_i$$. Alternative hypothesis sets that response $$y_i$$ follows this scheme: $$E(y_i|x_i)=\alpha+\beta x_i,(i=1,\ldots,m)$$ $$E(y_i|x_i)=\alpha+\beta x_i+\delta(x_i-z),(i=m+1,\ldots,n)$$ Where $$\alpha,\beta,\delta$$ are unknown. The goal of this formulation is showing that expected response given $$x$$ does not change at $$z$$ but the slope $$\beta$$ does. I have tried to arrange the expectations in normal equations but I got large chains of values and I do not know how to show this. Could you please help me how to adress this question.