Expectation of Gamma times S$^2$ in Black-Scholes model

Can somebody prove that:

$$E[S_t^2 \times \Gamma(t,S_t)] = S_0^2 \times \Gamma(0,S_0)$$

where $S_t$ follows a lognormal process as in the Black-Scholes model, and Gamma is the second derivative $\partial^2 C/\partial S^2$ of the option price with respect to S.

I can see it is true using simulation, but I can't prove it. It seems to be true for the Vega as well.