Let $G$ be a compact, connected Lie group acting smoothly on a compact, connected and oriented smooth manifold $M$. We denote by $H_G^*(M)$ the corresponding equivariant cohomology.
We have a canonical map, the characteristic map, $$ c:H^*(BG)\rightarrow H_G^*(M) $$ that endows $H_G^*(M)$ with the structure of an $H^*(BG)$-module.
There's also a canonical restriction map $$ r:H_G^*(M)\rightarrow H^*(M). $$ We say that $M$ is equivariantly formal if $r$ is onto.
Any hints on how to prove the following will be appreciated:
Propositon: If $M$ is equivariantly formal, then $H_G^*(M)\simeq H^*(M)\otimes H^*(BG)$ as $H^*(BG)$-modules.
N.B.: I am aware that there are many different ways to define equivariant formality, but I'd like to use only the given definitions, if possible.