We can now evaluate the left hand side, with the help of the original Ptolemy's theorem applied to the inscribed quadrilateral :

This probability approaches 1 as the total string approaches infinity, and thus the original theorem is correct.

Schottky's original theorem did not give an explicit bound for f. gave some weak explicit bounds.

The original theorems did not use the language of distributions, and instead applied to square-integrable functions.

Matsumoto's original theorem is even more general: For any root system, it gives a presentation for the unstable K-theory.

This result is analogous to the original Rice's theorem because both assert that a property is "decidable"

Idea of the proof: Apply the original theorem to the function sequence with the dominating function .

The original Rouché's theorem then follows by setting and .

The result initially stated is an intrinsic and global converse to the original theorem, therefore.

In the original theorem, the separating language was arbitrary.