Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_{\mathrm{t}}=y_{\mathrm{t}} \supset \cdot y_{\mathrm{t}}=z_{\mathrm{r}} \supset x_{\mathrm{r}}=z_{\mathrm{r}}$