Partial correlation — the residual-correlation identity

For the three random variables $X_1,X_2,X_3$ let $\hat X_1$ and $\hat X_2$ be the fitted values of $X_1$ and $X_2$ respectively after regressing on the linear function of $X_3$. The partial correlation coefficient between $X_1$ and $X_2$ after removing the effect of $X_3$ is the simple correlation coefficient between: