  We define $1$-cocycles on coideal $*$-subalgebras of CQG Hopf $\ast$-algebras
and consider the condition for $1$-cocycles to extend to $1$-cocycles on
Drinfeld double coideals. We construct a $1$-cocycle on a Podle\'{s} sphere,
which extends to a $1$-cocycle on $SL_q(2,\mathbb{R})$ and prove that it is
proper.
