There is also a very quick way to see both of these statements, if one is willing to assume that M-theory exists.
In that case, consider M-theory on a fibration with fiber the torus $S^1_A \times S^1_B$. As the radius of $S^1_A$ goes to zero this becomes 10d perturbative type IIA string theory. Then as also the radius of $S^1_B$ goes to zero and we T-dualize it, this becomes 10d perturbative IIB string theory. But the original M-theory did no require any of the two radii to be small, and so if T-duality has an M-theory lift as it should, then M-theory on any such torus should be non-perturbative IIB string theory. That's the picture of F-theory.
Now going through the behaviour of the coupling constants under T-duality (here) one finds that the complexified IIB coupling depends only on the conformal structure of the torus, not on its radii separately. In conclusion then the Moebius group $\mathrm{SL}(2,\mathbb{Z})$ of conformal transformations acting on the torus, which acts by 11d diffeomorphisms and hence ought to be a symmetry of M-theory, acts on IIB states. This is supposed to be the S-duality $IIB \overset{S}{\leftrightarrow} IIB$.
Next, consider the same story but with a $\mathbb{Z}_2 := \mathbb{Z}/2\mathbb{Z}$ orbifolding of the torus included. Take $\mathbb{Z}_2$ to act on $S^1$ by reflection on any line through the center of the circle (when thought of as embedded into $\mathbb{R}^2$ in the canonical way). Write $S^1 // \mathbb{Z}_2$ for the resulting global orbifold.
Now M-theory compactified
Both of these statements are originally due to Horava-Witten 95.
But as before, the symmetries of the torus act: Exchanging the two circle factors gives S-duality between heterotic E and type I' string theory. And T-dualizing, as before, turns this into a duality between HO and type I:
$$
\array{
HE
&\overset{KK/\mathbb{Z}^A_2}{\leftrightarrow}&
M
&\overset{KK/\mathbb{Z}^B_2}{\leftrightarrow}&
I'
\\
{T}\updownarrow
&& &&
\updownarrow {T}
\\
HO
&& \underset{\phantom{A}S\phantom{A}}{\leftrightarrow} &&
I
}
$$
This is what Horava-Witten 95 discuss around their p. 16.