Need help with subbase proof

[Cratylus]

I am using A First Course in Topology by Robert Conover
I am trying to prove the following
Theorem
The collection of subset of the plane
S={(a,b)X R,RX(c,d)|a<b,c<d} of all “strips” of the usual
topology of the plane

I am trying to show it is subbase

8.2 Theorem Let S be any collection of subsets of X Then S is a subbasis for a topology on X

My attempt

(a,b) XR={(x,y)|a<x<y} and R X(c,d)={(x,y)|c<y<d}
Both of these are open
Then by Theorem 8.2, since S is collection on set (a,b)XR and
RX(c,d) ,S is a subbase

I realize my proof sucks.Any help would be appreciated.