Theorem II1
The inversive crossratio is invariant under the Mobius group M(2).
Proof
Suppose that a Mobius function m is defined by
m(z) = (az+b)/(cz+d), with adbc ≠ 0.
Then we have
m(u,v,w,z) = (m(u)m(w))(m(v)m(z))/(m(u)m(z))(m(v)m(w)).
Considering each factor in turn:
m(u)m(w) 
= ((au+b)(cw+d)(cu+d)(aw+b))/(cu+d)(cw+d) 

= (adbc)(uw)/(cu+d)(cw+d), 
m(v)m(z) = (adbc)(vz)/(cv+d)(cz+d),
m(u)m(z) = (adbc)(uz)/(cu+d)(cz+d),
m(v)m(w) = (adbc)(vw)/(cv+d)(cw+d).
When we combine these, the (adbc) and the (c?+d) all cancel,
and the result follows at once.
