Homotopy invariance of de Rham cohomology
= Homotopy invariance of de Rham cohomology
Smoothly homotopic maps induce the same map on de Rham cohomology. Integrating <Cartan's magic formula> along a homotopy gives a cochain homotopy
$$
F_1^*-F_0^*=dK+Kd.
$$