$\displaystyle \frac{dy}{dt}F(y(t))=g(t),\int_{to}^{t}\frac{dy}{d t}F(y(t))=\int_{to}^{t}g(t)=F(y(t))-F(yo), \int_{t}^{to}f(r)dr=\int_{to}^{t}g(s)ds,$

in differential equations when solving an initial value problem for c, you have this proof in Braun where he changes f(t) to f(r), g(t) to g(s)- I'm not sure what the reasoning is, maybe you can use any arbitrary variable? the time dependence of y on t also remains?