|The Mirror Property
Suppose that L and M are distinct i-lines, then
Since L, M are distinct, M must contain points not on L.
Since inversion in L maps points on one side of L to points on the other side,
M must contain points on either side of L.
It follows that, if iL(M) = M, then M must cut L twice in E+.
Suppose that they meet in A and B.
As in the proof of the Uniqueness Theorem, we invert in a circle C with centre A.
Now consider the general case:
By the third part of the Algebraic Inversion Theorem,
iC° iL = iL'° iC,
|Algebraic Inversion Theorem|