Hybrid Sequence:
The starting point is $\lib{ss-real}$.
First, separate the cases for $\U$ into different branches of an if-statement. Within each branch, we can inline the behavior of $\Filter(\U,\cdot)$. These changes have no effect on the library's behavior.
$S_2$ is not used at all in the if-branch, so it can be removed without affecting the library's behavior.
The ``else if" branch generates and returns a OTP encryption of $\ptxt$, using $S_1$ as the key. We can use the security of OTP to show that the resulting output is uniform. In this three-hop maneuver, $S_1$ becomes $\key$ from the $\lib{otp-real}$ library and $S_2$ becomes $\ctxt$.
The branches of the if-statement can now be unified by writing them in terms of $\Filter$. The result is $\lib{ss-rand}$, which completes the proof.
$\lib{ss-real}$
$\ssshare$($\ptxt, \U$):
if $|\U| \ge 2$: return $\myerr$
//
$\Share(\ptxt)$
$\Filter\bigl(\{1\}, (S_1,S_2)\bigr)$:
$S_1 \gets \bits^\ell$
$S_2 $
${}:= {}$
$S_1 \oplus \ptxt$
$\otpenc(\ptxt)$
${}\gets \bits^\ell$
return
$\Filter(\U, S_1, S_2)$
$(\mynull, S_2)$
$\Filter(\U, S_1, S_2)$
$\link$
$\lib{otp-real}$
$\otpenc$($\ptxt$):
$\key \gets \bits^\ell$
$\ctxt := \key \oplus \ptxt$
return $\ctxt$
$\lib{otp-rand}$
$\otpenc$($\ptxt$):
$\ctxt \gets \bits^\ell$
return $\ctxt$