  We define the $C^*$-algebra of quantum real projective space $\R P_q^2$,
classify its irreducible representations and compute its $K$-theory. We also
show that the $q$-disc of Klimek-Lesniewski can be obtained as a non-Galois
$\Z_2$-quotient of the equator Podle\'s quantum sphere. On the way, we provide
the Cartesian coordinates for all Podle\'s quantum spheres and determine an
explicit form of isomorphisms between the $C^*$-algebras of the equilateral
spheres and the $C^*$-algebra of the equator one.
