Let $\r$ and $\s$ be ordinals with the order topologies. 

It is known from \cite{KTY} that metacompactness, 

screenability and weak submetaLindel\" ofness are 

equivalent for every subspace of $\r\times\s$. 

However there are a metacompact subspace of 

$(\wone+1)\times(\w_2+1)$ which is not subparacompact, 

and a subparacompact subspace of $(\w+1)\times(\wone+1)$ 

which is not paracompact, see \cite{KTY, Example 4.2 and 4.4}. 

Moreover it is not difficult to show that these examples 

are not paraLindel\" of.  

So it is natural to ask whether all paraLindel\" of 

subspaces of $\r\times\s$ are paracompact for every 

ordinals $\r$ and $\s$. 

In this paper, we will see that paraLindel\" of subspaces 

of $(\rho+1)\times(\wone\cd\w)$ are paracompact 

for every ordinal $\r$, where $\wone\cd\w$ denotes 

the ordinal number $\wone+\wone+\cd\cd\cd$($\w$-times), 

see \cite{Ku, I Definition 7.19}. Moreover we will show 

that paraLindel\" of subspaces of $(\ro+1)^2$ are paracompact 

for every ordinal $\r<\wone\cd\wone$. 

And we will construct a non-paracompact subspace $X$ of 

$(\wone\cd\wone)\times(\wone\cd\o+1)$ which can be represented 

as the locally countable union of clopen paracompact subspaces.