A deep theorem of Behnke and Stein (1948) asserts that X is a Stein manifold.

It is a deep theorem of Federer-Fleming that the boundary is then also an integral current.

It is a deep theorem of Ratliff that the converse is also true.

On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics.

A canonical form may simply be a convention, or a deep theorem.

By contrast, the existence of Jordan canonical form for a matrix is a deep theorem.

In addition to its unifying approach, universal algebra also gives deep theorems and important examples and counterexamples.

It is a deep theorem of Grothendieck that a Hilbert scheme exists at all.

Even the fact (if true) that a finite number of extra digits will ultimately suffice may be a deep theorem.

Many deep theorems on the structure of finite groups use characters of modular representations.