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[[File:Illustration of Riemann Mapping Theorem.JPG|Illustration of Riemann Mapping Theorem]]
==Elaboration==
In the above figure, consider ''D''{{su|b=1}} and ''D''{{su|b=2}} as two simply connected regions different from ℂ. The [[Riemann mapping theorem]] provides the existence of ''w=f(z)'' mapping ''D''{{su|b=1}} onto the unit disk and existence of ''w=g(z)'' mapping ''D''{{su|b=2}} onto the unit disk. Thus ''g''{{su|p=-1}}''f'' is a one-one mapping of ''D''{{su|b=1}} onto ''D''{{su|b=2}}.
If we can show that ''g''{{su|p=-1}}, and consequently the composition, is analytic, we then have a conformal mapping of ''D''{{su|b=1}} onto ''D''{{su|b=2}}, proving "any two simply connected regions different from the whole plane ℂ can be mapped conformally onto each other."
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