Prove that for all $\displaystyle \alpha$ in $\displaystyle \omega_1$,

$\displaystyle \text{seg}_{\omega_1}(\alpha) +_o \omega_1 =_o \omega_1$

and

$\displaystyle
\text{seg}_{\omega_1}(\alpha) \cdot_o \omega_1 =_o \omega_1$.

Notation: $\displaystyle \omega_1$ denotes the first uncountable ordinal. $\displaystyle \text{seg}$ above denotes the initial segment, $\displaystyle \cdot_o$ denotes the product of posets, $\displaystyle +_o$ denotes the sum of posets. $\displaystyle U <_o V \Leftrightarrow (\exists x \in V ) [U =_o \text{seg}_V(x)]$. $\displaystyle =_o$ denotes order isomorphic.