Möbius transformations mapping one circle to another with two prescribed points

Let C and \(C ^ { \prime }\) be two circles and let \(z _ { 1 } \in C , z _ { 2 } \notin C , z _ { 1 } ^ { \prime } \in C ^ { \prime } , z _ { 2 } ^ { \prime } \notin C ^ { \prime }\) . Show that there is a unique fractional linear transformation f with \(f ( C ) = C ^ { \prime }\) and \(f ( z _ { 1 } ) = z _ { 1 } ^ { \prime } , f ( z _ { 2 } ) = z _ { 2 } ^ { \prime }\)