The Fundamental Theorem of Inversive Geometry If L = (α,β,γ) and L' = (α',β',γ') are lists of distinct points of E+, then there is a unique t in I+(2) which maps L to L'.
Proof By the (0,1,∞) Theorem, there exist elements r, s of I+(2) such that r maps L to (0,1,∞), and s maps M to (0,1,∞). Then t = s-1or maps L to M. Suppose that u also maps L to M. Then sou maps L to (0,1,∞). By the uniqueness clause of the (0,1,∞) Theorem, only r maps L to (0,1,∞). Thus sou = r, so u = s-1or = t, i.e. t is unique.

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