proof of Theorem II1

Theorem II1
The inversive cross-ratio is invariant under the Mobius group M(2).

Suppose that a Mobius function m is defined by
m(z) = (az+b)/(cz+d), with ad-bc ≠ 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)
= (ad-bc)(u-w)/(cu+d)(cw+d),

m(v)-m(z) = (ad-bc)(v-z)/(cv+d)(cz+d),
m(u)-m(z) = (ad-bc)(u-z)/(cu+d)(cz+d),
m(v)-m(w) = (ad-bc)(v-w)/(cv+d)(cw+d).

When we combine these, the (ad-bc) and the (c?+d) all cancel,
and the result follows at once.

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