Segments, posets

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

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

and

$<br /> \text{seg}_{\omega_1}(\alpha) \cdot_o \omega_1 =_o \omega_1$ .

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